A friend of mine who runs an AD&D game online recently got into a 5E game and decided to play a Wizard as his first character. And he was pretty confused. There are spells in your spell book, spells you prepare, spell slots, cantrips that are at-will, and ritual spells that are sort of at-will, but take time.
So he asked me to explain it to him since his DM couldn't in a way that made sense to him.
Honestly, I flailed around at this for a bit, even though I get 5E and how it runs spell casters.
The way I finally explained it was to break down for the first three levels how many spells you get in the spellbook, how many of those spells you prepare each day, and then how many spell slots you have to cast prepared spells. Plus, then reminding him of ritual spells and how they work. (He had cantrips down, they're easy.)
I was more wordy than this, but it broke down along these lines:
At 1st level, you have 6 spells in your spellbook. You prepare any 4 of them (assuming a +3 Int bonus at 1st level). You can cast twice. That may be two different spells you prepared, or the same spell twice.
At 2nd level, you get two more spells in your spellbook. You prepare any 5 of the 8 spells. You can cast three times, in any combination of the 5 prepared spells.
At 3rd level, you get two more spells in your spellbook, probably but not necessarily 2nd level ones. You can prepare any 6 of your 10 spells. You can cast six times, four times MUST be 1st level spells, and two times could be 1st or 2nd level spells, in any combination of your 6 prepared spells.
And then you can cast any ritual spell without spending a spell slot as long as you have the time to cast it, but since they're mainly out of combat spells, you usually will.
While the flexibility of this is nice (utility spells don't need to be ignored and never prepared until after the need for them is known), it is fairly complex.
Preparing X spells of each spell level per day, and then just casting those spells old school style is just so much simpler.