Xanathar’s Guide to Everything answers almost all your issues.
Before I tackle the specific questions, I’ll answer the overarching question.
What does the Herbalism Kit actually do?
This is clarified in Xanathar’s Guide to Everything. Essentially, most of what the Herbalism Kit does is act as a supplement to other abilities. The examples given in Xanathar’s Guide are using your knowledge of nature to assist with Arcana checks relating to plants and potions, assisting with Investigation checks relating to looking through anywhere with overgrown plants, assisting with Medicine checks when using herbal remedies, and so on.
Most of the kits in the game don’t have an obvious list of things to do with them, and work the same. They are mostly roleplay opportunities for your character to shine in a specific way. If you’re stuck in a forest and need to determine which plant is which, you would mention to your DM that you have expertise in plants due to the Herbalism Kit, and likely would be given advantage or proficiency bonus on the roll, if not just be told outright immediately.
That means (Herbalism Kit) has a limit usages?
No. When you craft an item, the items that are used to craft it are factored into the cost.
In addition to the appropriate tools for the item to be crafted, a character needs raw materials worth half the item’s selling cost. (XGE, 128)
So yes, while you technically use some of the kit to create the item, you simply replace the parts that you used afterwards.
How can I craft healing herbs, remedies or healing potions? (Anything that restores HP)
Page 130 of Xanathar’s Guide to Everything covers the specifics of Potion of Healing crafting. In summary, a basic Potion of Healing takes a day of downtime, 25gp of materials, and proficiency in (and access to) the herbalism kit.
How can I gather the resources to craft?
Technically speaking, you don’t. Crafting an item in 5th edition handwaves materials and resources by simply charging an amount of gold depending on what it is you are crafting and just says “you buy what you need”.
Your DM may rule otherwise, and were I the DM of your game and you wished to spend time gathering the herbs needed rather than simply buying them, I’d happily allow that as a roleplay opportunity. You spend more time doing it, but need to spend less gold. However, that’s up to your DM, and would be homebrew rules.
How can I identify or apply herbs? If I identify an herb, can I gather it? How much time it takes that?
This is another situation where it is up to your DM. Your DM will need to present you with the opportunity to tell what type of herb the given plant is, and you would mention to your DM that you have Herbalism Kit proficiency so you would be able to identify it. You can also mention it on your own, as well; if you are in an area with potentially valuable/helpful herbs or plants, ask your DM if your character could search for and gather something useful.
While not part of your question, there’s a line in your question that is concerning and should be mentioned to a first-time player as you mention you are.
I don’t want to ask to the DM how it works because I’ve already asked him several questions and I don’t want to bother him.
The thing with D&D is that the DM is the one who runs the game. The rulebooks are secondary to how they want to run things. If they’ve agreed to be DM, then they’ve agreed to also answer rules questions when they come up. To do otherwise is just plain silly; he’s the DM. That’s the job he accepted when he chose to become DM. If he doesn’t want to answer rules questions from his players, he shouldn’t be DM.
Feel free to bring up the rules I’ve listed here, and he very well may just say “okay, that’s how it works, whatever”. But to be afraid of asking your DM questions about the rules in the game he’s running is going to cause problems later on, and you should be able to have open dialogue with him/her.