Nowadays people often say "way more", "way better" etc. instead of using the word "much". How did this become popular usage?
Etymonline offers this commentary for the use of way as an adverb
c.1200, short for away (adv.). Many expressions involving this are modern and American English colloquial, such as way-out "far off;" way back "a long time ago" (1887); way off "quite wrong" (1892). Any or all of these might have led to the slang adverbial meaning "very, extremely," attested by 1984 (as in way cool).
The forms "way more," "way better," and the like, in the sense of "far more," "far better," and so on, may go even farther back than Etymonline's dating (cited in bib's answer) suggests.
Search results for 10 phrases
I ran Google Books searches for "is way above," "is way back," "is way behind," "is way better," "is way far," "is way more, "is way off," and "is way out," "is way too," and "is way up" for the period 1800–1950, with the following earliest-occurrence matches for each.
From Testimony Taken Before the Canal Investigating Commission, sixth session (June 8, 1875), in Documents of the Senate of the State of New York, Volume 4 (1876):
Q. How many chains is it between those two locks, 18 and 19?
A. I don't know; lock 19 is way above the lower Mohawk.
From Thomas Norwood, Plutocracy: Or, American White Slavery (1888):
"Second : the dead-line for a girl is farther back, higher up on this plane, than the dead-line for boys. It is way back beyond where you met the procession, Mr. Woolhat. " Third : When a girl once slips over that dead-line, both men and women drive her on, down, farther down, until she falls over into that pit, from which she can never rise.
From "From the Pacific Coast," in The Clothier & Furnisher (May 9, 1889):
In the retail furnishings trade there is a noticeable improvement. A full month of as perfect weather as any reasonable being could ask for, has, of course, had a most beneficial effect on this line of goods. Even with this improvement they are making the same complaint as at this time last month to the effect that tho season is way behind, and that they cannot now make up for lost time.
From Harold Bindloss, Masters of the Wheat-Lands (1910):
"Then," Sally responded, "it is way better that you didn't marry him." She paused, and seemed to search for words with which to express herself. " I knew all along all there was to know about Gregory—except that he was going to marry you, and it was some time before I heard that—and I was ready to take him. I was fond of him."
From the entry for December 26, 1888 in With Walt Whitman in Camden, volume 1 (1914):
I was speaking of Emerson and W[hitman] as the giants of our time in America—the only giants. "I assume that you stand incomparably higher than all others." I said: "I think less of Bryant than probably you do." "I have an idea you do : I think a great deal of him : regard him highly : but I clearly perceive that you are essentially right — that taken all in all Emerson is way, far, above all others: not one to share his glory."
From U.S. Congress, Hearings of the House Select Committee to Investigate the Interstate Migration of Destitute Families (1940) [combined snippets]:
Mr COOK. I kept going deeper and deeper from that day on, and then in 1934, when that drought hit us, there was no feed to be gotten, you just had to buy what you could get, and we paid as high as $21 for a load of oat straw.
Mr. PARSONS. Yes.
Mr. COOK. $21 a ton, which is way more than straw ever has been worth or ever will be. Well, then, after hay got way up there and lots of the alfalfa was pretty risky to buy, because they'd water it to keep the leaves on, and watered it too heavy. I have seen alfalfa that when a man went to cut the wires on the bale you couldn't get it loose with a crowbar.
From the address of Mr. Green of Tennessee Conference (May 26, 1844), in Debates of the General Conference, of the Methodist Episcopal Church, May, 1844 (1845):
Methodism has but one heart. Say you are sorry that Bishop Andrew is connected with slavery, but that you have no constitutional power to touch it, as he has violated no rule of discipline. Some of you can do this, you have told me so, that this would enable you to satisfy all that you know ; but the great difficulty is way off yonder, somebody has told you. Will you then cut down this beautiful tree because you hear it reported that some of the distant branches will fall off if you do not cut the tree down. Let me entreat you to do all you can to save the church.
From Arthur Morecamp, Live Boys: or Charley and Nasho in Texas (1878):
I ran into Auntie's room so out of breath I could hardly speak. "Oh, Uncle! the dogs have treed a great big panther, and Nasho is watching him : wont you take your gun and shoot him — quick Uncle, please, before he gets away. He run up the tree we was standing against and is way out on a top limb. Make haste Uncle, please.”
From Alfred Lewis, Wolfville (1897):
'Pass over them documents for Cherokee Hall, an' don't hold out nothin' onto us. We-alls is 'way too peevish to stand any offishul gaieties to-day.'
From Massachusetts State Board of Agriculture, Annual Report of the Secretary of the Massachusetts State Board of Abriculture (1877):
The rye is looking so finely, and I have so much confidence in my ability to to raise corn with these commercial fertilizers, that I have concluded to fence that piece of ground, and, in addition, another piece which I have ploughed up, and I propose to plant corn next year and manure it with commercial fertilizers, that I have concluded to fence that piece of ground, and, in addition, another piece which I have ploughed up, and I propose to plant corn next year, and manure it with chemical fertilizers, because it is way up on a hill,—although it is pretty flt when you get there,—so that you cannot haul manure up there, but you can take the fertilizers up on your back.
Conclusions from these results
The years of first occurrence for these ten expressions are, from earliest to latest, 1844 (way off), 1875 (way above), 1877 (way up), 1878 (way out), 1888 (way back), 1888 (way far), 1889 (way behind), 1897 (way too), 1910 (way better), and 1940 (way more). It seems safe to say that by the end of the 1870s, colloquial wordings in which way functioned as a synonym for far or much were well established in U.S. speech.
To generate a manageable number of results, I made is the first word of each search phrase, greatly reducing the number of matches for each phrase. It seems highly likely that even earlier matches for way used in the sense of far or much exist, in situations where way is not preceded by is.
The search results also seem to confirm Etymonline's remark that way, when used in this sense, is "short for away": On several occasions, early matches for the search phrase rendered way as the contracted form 'way of away.
It appears that using way in place of far or much has been part of colloquial U.S. speech for 140 years or so, with one instance going back 170 years. The usage may be more frequent now than in the past, but it has been around for very long time.