The only times the keeper may draw mythos or trauma cards is (1) during keeper setup (at the end of setup, under step 6 of keeper setup, "Starting Cards and Threat"), it tells you how many of each type of card, mythos and/or trauma, if any, to start out with; or (2) by paying threat to activate a keeper action during the keeper action phase (the keeper action cards are called "Evil Presence" and "Dark Ritual"- you will only use 1 of these keeper actions, depending on the scenario you are playing); or (3) if an event instructs the keeper to do so (for example, under 1B of event IV of the "Green-eyed Boy" scenario it instructs the keeper to draw 3 mythos cards. You cannot freely draw mythos or trauma cards. You must wait for the game to instruct you to draw them or you must pay threat to draw them, according to your specific keeper action card.
Once you have drawn mythos or trauma cards, you keep them in your hand until you play them. Once they've been played, they are discarded and never seen again until your mythos or trauma draw decks are empty and you need to reshuffle your mythos or trauma discard piles. If you DON'T play mythos or trauma cards they STAY in your hand until you have 4 of each in your hand. This means you may have a total of 8 cards in your hand, 4 mythos and 4 trauma. If you ever get more than 4 mythos and/or 4 trauma, you must discard down to 4 of that card type. You may never have more than 4 mythos cards in your hand and you may never have more than 4 trauma cards in your hand.
You may only play mythos cards on an investigator DURING their turn, never during the keeper's turn. If a mythos card has a room requirement listed on it, then you may only play it, during an investigator's turn, if they start in that room or move into it during their movement or action phase.
You may only play trauma cards IMMEDIATELY after an investigator has taken damage or horror. If the trauma card has a number listed in the upper right corner of the card, then you may only play the card IMMEDIATELY after an investigator has taken damage or horror AND has remaining health or sanity equal to or less than the number shown.
All of these rules are explained on page 11, 16, and 25 of the rulebook.