There's no "closest descendant" of PIE, just as there's no "closest descendant" of the last common ancestor of modern apes--each extant descendant has been developing just as long as any other. You will sometimes hear it claimed that Lithuanian is the "closest" modern language to PIE--the Indo-Europeanist Antoine Meillet is attributed with saying that "anyone wishing to hear how Indo-Europeans spoke should come and listen to a Lithuanian peasant," and that sentiment gets a lot of play with Lithuanian nationalists. In the intervening century, we've come to learn a lot more about Indo-European, and while Lithuanian might well be the most conservative (least-changed) Indo-European language still living, at least by certain metrics, it has some significant differences from Indo-European. Another language that gets a lot of play in nationalist circles as the "closest" to Proto-Indo-European is Sanskrit. It is indeed old, though predated in written form by Hittite, Mycenean Greek, Ancient Greek, Latin, and various Prakrits. Sanskrit has an impressive oral tradition stretching back very many centuries, and has quite a lot in common with Proto-Indo-European. That said, it too has a lot of differences from PIE: for instance, 3 vowels that were clearly distinct in PIE are collapsed into one in Sanskrit, and Sanskrit also adopted a whole new class of consonants from Dravidian languages indigenous to the Indian subcontinent.

The consensus among linguists is that PIE developed not near the modern Baltic states, but a few hundred miles to the southeast, on the steppe north of the Black and Caspian seas, roughly 5500 years ago. The question of how it spread is more contentious, and /u/brigantus or others more conversant with migration theory will know the subject better than I do.