DnD 5e - Tools and Gaming Kits
Users with proficiency in a set of tools or a game set can add their proficiency bonus to checks made with that set of tools or that gaming set. Different tools have different functions, and in some cases they can be used for craftinng items during downtime (see Downtime, earlier in this guide) or for other purposes closely related to adventuring. The exact contents of most of these kits is left to your imagination.
For the table listing tools and gaming sets, including their prices, see page 154 of the Player's Handbook. For more information on what you can do with specific tools, see pages 78-85 of Xanathar's Guide to Everything.
The Player's Handbook lists 17 types of artisan's tools, and you could imagine more if you gave them some thought. The individual types aren't detailed beyond their name in the Player's Handbook, but Xanathar's Guide to Everything explores each in detail, even suggesting some example uses for each set of tools.
Beyond the options listed in Xanathar's Guide to Everything, the only use for artisan tools that has published rules is downtime activities. You can craft items using appropriate tools, or you can Work by magic an intelligence check with the tools. Your Dungeon Master might try to incorporate your tool proficiencies into the game, but it's hard to use calligrapher's supplies in a dungeon.
Disguise kits are basically a full prosthetic makeup kit. You can use it to create visual disguises. The Player's Handbook provides little information on what exactly you can do, but Xanathar's Guide to everything goes into detail.
Unsurprisingly, a Forgery Kit is used to forge documents. This doesn't come up much in a typical game of Dungeons and Dragons, but if you're doing a lot of political intrigue, court politics, or general crime which may involved documents like party invites, land deeds, pasports, etc. you might enjoy this.
A "gaming set" is strangely restricted to the physical pieces used to play a game. The Player's Handbook lists four examples: a dice set, a dragonchess set, a playing card set, and a three-dragon ante set. A dice set likely represents conventional six-sided dice, so you can play any game that uses only dice. A playing card set similarly represents an equivalent of a real-world deck of playing cards, allowing you to play poker, blackjack, etc.
Conversely, Dragonchess and Three-Dragon Ante are individual games (both of which have real-world rules) with specific pieces or cards which aren't useful for other games. Selecting them over "dice set" or "playing card set" is like choosing to know how to play Monopoly over knowing how to to play every card game in existence which can be played with a standard deck of playing cards.
I don't understand why it works this way. If it bothers you as much as it bothers me, talk to your DM about using a house rule to change it.
Herbalism kits serve primarily to allow you to craft Antitoxins and Potions of Healing, both of which are fantastically useful for adventurers.
Player characters most commonly use musical instruments when playing a bard because bards can use them as a spellcasting focus (see magic, later in this guide). You can also use a Charisma check with a musical instrument to Work during downtime.
Used for navigating at sea, and literally nothing else.
Primarily used to craft, handle, and apply poisons. Poison is a terrible option for player characters unfortunately. The only published poison which you can freely buy is "Basic Poison", listed under Adventuring Equipment. It costs 100gp to do a pitifully small amount of damage which can be resisted with a DC 10 constitution saving throw.
Easily the most important set of tools for adventurers, Thieves' Tools allow you to add your proficiency when disarming traps and opening locks. Adventurers tend to encounter both of those frequently. I strongly recommend that every party have at least one member who is proficient in Thieves' Tools.
Vehicles (Land or Water)
I honestly don't know why this is listed under tools in the Player's Handbook. Mounts and Vehicles is literally on the next page.