As a general rule, the answer is token cards and dice.
Objects that don't physically exist in Magic tend to be token permanents, copies of spells on the stack, and abilities on the stack.
Tokens are often easy. Wizards produce token cards for non-copy token permanents within a set, so you can use these. If you don't have the appropriate token, you can use reminder tokens (e.g. I have 6/6 "Wurm" tokens for a deck that needs 1/1 "Worm" tokens), other objects, or relevant reminder cards from outside of the game. If your deck is sleeved, you can use unsleeved cards face-down. Dice are helpful as well - I can put a d6 on my face-down card to represent the power/toughness of the construct it represents.
Occasionally effects need to be remembered that aren't stored by the game on its own. Some examples of this are renowned/monstrous, "gain control" effects, or storm counts. Again, dice are helpful. At this point I have dice in a variety of colours. Simple +1/+1 counters will always be on white dice, but a renowned or monstrous creature will have its +1/+1 counters on a translucent coloured die.
If I've created tons of tokens for which I don't have the card, things can get more complicated. I might need dice to indicate the P/T of a group of creatures, the number of them, and additional counters on them. Again, different shapes/colours of dice can help you develop a consistent standard. I use d10s/d% to track numbers of objects, d6s for counters on an object, and d20s for life.
Copies on the stack tend to be of other things on the stack or of something held in exile, so you can refer to it. The stack doesn't usually get too deep, so it's not difficult to resolve it without forgetting what things are there. If it gets complicated you can put the exiled cards as reminders in a physical stack.
The reason it doesn't usually get too complicated is because you'll usually be able to refer to the effects that created stuff, and that there won't usually be too many different kinds of things at once. If I can't remember what your token creatures are, I can look at your "Raise the Alarm" in your graveyard.
Lasting effects can be tougher to remember as the board doesn't indicate the state for you. For example, if I play an instant or sorcery to gain permanent control of your creature, it's easy to remember because your card is on my side of the board. If you then gain control of it temporarily, we have to remember to give it back to me afterwards. I'll put reminder objects (rubber bands, bottle caps, etc) on relevant cards (e.g. exile at end of turn, or don't untap). I know people who have other objects (e.g. coins) that they put on their library to remind them to do something during upkeep, before drawing.
As you notice, Magic does have these moments where the board state can be complicated and not self-preserving. In general, players don't worry too much about it. As long as you've got techniques (as I mentioned, tokens and dice) to remember the more common details, you can improvise ways to deal with other situations. Most of the time you'll remember what's going on, and sometimes you'll forget but it won't matter too much. As you become more experienced with the game, you'll tend to remember details better and you'll know what you can infer from the board state. If you're playing in a deck or format where certain details matter, you'll choose ways to keep track of them.
If you come to a situation where these techniques are overcomplicated, insufficient or just don't work for you, you can write down notes as needed.
