A berth refers to the distance that a ship or boat should bear off of ships, rocks, and other obstacles, as well as the space that should be left when a ship anchors and bears from (lies off of) a point.
For instance, Henry Mainwaring, in the Nomenclator Navalis (1620-23), defines a berth as
a Convenient distance & Roome to Moore a shipp in Alsoe when they would goe cleere off a Poynt or a Rocke, They saye take a good Berth That is goe a pretty distance of to seaboard of it
As the dictionary entry you cite shows, to bear off means to move away from land. That's exactly the action Mainwaring describes as a good berth: going to seaboard. Berth also qualifies the place a ship would moor or anchor itself - a ship needs to be able to bear off, so the berth is the space to do that.
Early use sometimes preserves the use of off to qualify berth. The earliest text I can find using berth, from A true discourse of the late voyages of discouerie, for the finding of a passage to Cathaya, by the Northvveast, vnder the conduct of Martin Frobisher Generall by George Best (1578), features berth off as well as berth of:
And now the whole Fléete plyed off to Seaward, re∣soluing there to abide, vntill the Sunne might consume (or the force of wind disperse) these Ise from the place of theyr passage: and being a good berth off the shore, they toke in their Sayles, and lay adrift.
Some of the Shippes, where they could find a place more cleare of Ise, and get a litle berth of sea roome, did take in their Sayles, and there lay adrift.
And here is Purchas his Pilgrimes, Samuel Purchas (1625):
Note, that this land of Camboia lyeth more Easterly in our plats, then it should, for wee find South South-west to goe alongst the land a faire berth off.
One other example shows how early usage could use prepositional phrases to define berth relative to another position or direction. This is John Smith in An accidence or The path-way to experience (1626):
Watch bee vigilant to keepe your berth to wind∣ward: and that wee loose him not in the night.
Mooring and berth may also be linked through the verb "bear, v.1", meaning 39, which first emerged at the end of the sixteenth century:
- Chiefly Nautical: To lie off in a certain direction from a given point or place. (Cf. bearing n.)
1594 T. Blundeville Exercises vii. xxiv. f. 321 The Shipmaister knowing..how the port..beareth from the place from which he departeth.
So the source of berth could be this meaning of bearing from (lying off a point), bearing off (moving off shore), or an intermingled sense of both. The OED is making an educated guess.
One other wrinkle is that we can't entirely rule out the coincidence of the noun and verb forms, berth and to berth. The OED supposes that the verb to berth comes from the noun, and cites a first entry in the 1670s, 50 years after noun berth. However, verb forms berth and berthed appear from 1621 in Algiers voyage by John Button. Note the mingled uses of verb and noun forms:
The 21. at 6. in the afternoone we came to an-Anchor in Algiere Roade, order being first giuen by our Admirall how euery ship should berth her selfe: the manner followeth; first the Kings ships and the Golden Phenix came to an anchor at their place vpon a South and Northline, the Admirall riding in the middest of them, East of the South end of the Mould. Next to the Admi∣rall on the North side came to an anchor the Re∣formation, the Golden Phenix on the South, the Conuertine on the Southside, the Phenixe and Anthelope on the Northside the Reformation, the Vice-admirall innermost on the Southside, the Rere-admirall vttermost on the North side.
The kings ships hauing berthed themselues, the rest of the fleete who this time plyed vp and downe vnder sayle, came to a Sterne to the Ad∣mirals of their squadrons, giuing good berthes one to another, the winde being then Westerly: but if the winde had beene Easterly, then were they to haue beene anchored a head between the Kings ships and the Easterne shore, keeping the Kings ships and the Phenix betweene them and the towne, where the force of the Pyrates re∣mained.
So the verb also appears earlier than the OED attests.