May HIS camp become desolate, and let there be no one to dwell in IT

Acts 1:

15 In those days Peter stood up among the brothers (the company of persons was in all about 120) and said, 16“Brothers, the Scripture had to be fulfilled, which the Holy Spirit spoke beforehand by the mouth of David concerning Judas, who became a guide to those who arrested Jesus.

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Peter's mindset was that the Holy Spirit inspired David to prophesy about Judas in the Psalms.

20 “For it is written in the Book of Psalms, “‘May his camp become desolate, and let there be no one to dwell in it’;

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i.e., Psalm 69:25

and “‘Let another take his office’

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i.e., Psalm 109:8b.

What made him think that David was talking about Judas? Did Peter pick and choose and quote these verses out of context?

Yes, by our modern standard of scholarship. Psalm 69:

25 May their camp be a desolation; let no one dwell in their tents.

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In fact, the original was plural, not singular as quoted by Peter. He also changed the wording of the original. Jewish methods of hermeneutics allowed him to do it.