I have seen that it means "it is your turn". But is it supposed to mean something like "I touch to you"? What does "te toca" conjugate from anyway?
The verb tocar does not necessarily mean "to touch". It has a plethora of other meanings, as you can see in its definition in the DLE. Example:
Yo toco el piano. I play the piano. (3rd meaning.)
In fact, it is usual to joke with "yo sé tocar el piano" (I know how to touch/play the piano). The usual meaning would be with "play" and then you go and start touching the piano and say "¿ves?, sé tocarlo".
But there are also many other meanings which do not have to do with "touch" or "play". You can also say:
Tocamos a dos euros cada uno. We can share two euros per person. (23rd meaning.)
In the case of the sentence te toca a ti as in "it's your turn", there is no touching there at all. In fact we already had a debate about if it's the 19th meaning or the 21st meaning of the verb tocar.
- intr. Pertenecer por algún derecho o título.
- intr. Ser de la obligación o cargo de alguien.
So the proper literal translation of "te toca a ti" should be "the turn belongs to you". As for "te toca", you are just using the personal pronoun "te" as the direct object of the verb. You could drop "a ti" in that sentence as it is redundant, but it can also be included for emphasis.