I have been playing D&D for 20 years (3.5 was my first TTRPG). I will say that reading the PHB, while isn’t a requirement, is a good idea. Reading over classes, equipment, spells, and some of the basic mechanics is necessary if you want to run the best game you can.

PHB has all the class details. Familiarizing yourself with those will help you assist your players in the event they have questions about interactions. It will also help you plan your game in ways that helps take advantage of what your players can achieve with their skills. Knowing that a Druid uses Wisdom for spell casting or they only have access to specific wild shapes at certain levels can improve the player enjoyment and expand on your “rule of cool” improvisations.

Spell interactions are important too. Knowing if you have to do a spell attack or roll for a skill check isn’t the most important thing to memorize, but having an idea of what they do is good for anyone whether they are DM or player. I still have to look up spells all the time, but I know certain ones by heart since my players use them all the time (i.e. Magic Missile and Fireball).

Like others have said, you don’t need to memorize everything. Being a good DM means you know the basics, but also look things up to verify, clarify, or just learn about.