MTG how hellion crucible and cards like it work?

It's just colourless mana. It goes in your mana pool and waits for you to spend it. At the end of each step and phase, if you didn't spend it, it disappears. Just like any other mana. It does NOT mean you get to search for a land. Lands and mana are two different things - lands produce mana.

Wow, somebody's confused about what mana means and what land does. No, you don't search when a card says "Tap to add {1} to your mana pool" but just add another mana, which you use normally except for being colorless, of course.

And like all mana, it empties from your mana pool at the end of each phase unless you have a card like Upwelling in play.

If you are supposed to search for a land, the card will say it, like:

Wooded Foothills: Tap, Pay 1 life, Sacrifice Wooded Foothills: Search your library for a Mountain or Forest card and put it onto the battlefield. Then shuffle your library.

Where this idea that "Tap: Mana Symbol" means you search for a land came from, I don't know, but it's wrong.

You tap to add one colorless mana to your mana pool. MANA DOES NOT EQUAL LANDS. Mana is used to pay for costs and you can gain mana by tapping lands but there are other ways to gain mana.

Colorless mana can be paid for by any color mana or by colorless mana but colorless mana can only be used to pay for colorless costs.

Mana drains from the mana pool at the end of every step and phase.