Hang on, slow down here!
There are two different matters being questioned here.
1) Why is "a lot" preceded by is
A lot in this sense describes a certain group of something, which is singular. There are many other similar words, examples:
There is a lot of people here.
There is a team of people working on this project.
There is a group of bystanders watching the accident.
2) Why is "there are" sometimes shortened to "there's"
Yes, this actually does happen. I'm not claiming it's grammatical, but people do it. Simply for the reason that you cannot shorten "there are" to "there're." In that case, you can say "there's" when meaning "there are", but this can be used only in spoken or informal English. Examples:
There are many apples on the table. => There's many apples on the table.
For further reference see Is “there're” (similar to “there's”) a correct contraction?, where the currently accepted answer states:
A huge number of English speakers, even those that are well-educated, use there's universally, regardless of the number of the noun in question, so you will probably not receive any odd looks for saying or writing there's, and if you do, just cite the fact that it can't be incorrect if a majority of people use it.