How do players "see" several moves ahead? How can you predict what your opponent will play in the next 5 turns, 7 turns, 10 turns?

If the game is like a tree, then the more the turns the more it will branch out.

How can a player accurately predict what their oppoent will play several moves into the future when the number of possible combinations of moves to get ten moves ahead can be ridiculously large?

Please give a detailed explanation.

How do I learn/acquire this skill?
The answer is that you only consider the most sensible and most critical options. This prunes the tree down to manageable proportions.

One easy example would be the fact that retreats are usually not threatening and therefore less likely to be critical. (Of course the downside is that actually dangerous retreats are often overlooked.)

To learn how to recognise potentially critical moves you have to play through master games, solve puzzles and play games. Over time your brain will store typical moves and manoeuvres in certain position. This ability is called pattern recognition. The more patterns your brain stores the stronger its intuitive move suggestions will be, you will be able to prune the tree more.

Another aspect is called "forcing moves". A forcing move is a move that requires a certain response, i.e. severely restricts the opponent's choices. For example checks, threats of checkmate and other kinds of strong threats. If your moves are very forcing the tree will resemble a path, because every step from this path is either illegal or results quickly in very bad things. If GMs calculate very long variations, these usually contain very forcing moves. In positions without forcing moves even the best players don't look very far ahead.

There is also a non-intuitve conscious part to "seeing ahead". Even if your brain provides you with only the most likely good moves, you still have to visualise the outcome of every variation in your head. That requires training as well, for example solving studies or tactical puzzles with long variations.