how does this connect to a cellphone in order to have better signal and are there other parts that goes with it ?

This is a part of a signal boosting system - It cannot in itself help improve reception. It is included in zBoost signal amplifier models stated in the description. You must purchase the main signal booster system that already includes this item, if you do not have that already yet. 

Will it work on a ZB540 booster

Yes, this antenna will work with the ZB540 booster.

I am looking for a replacement antenna for my zboost. I love the product and everything else works but this is cracked. Could you suggest another that would work for zboost 545.

You would be able to use either omni-directional or uni-directional antenna shown at links below.

Omni antennas don't have to be pointed to nearest cell tower because they attract signals from all directions. This is similar to the original 75 Ohm antenna that came with this kit. Here's the link to the recommended replacement omnidirectional antenna:
https://www.signalbooster.com/products/omni-short-antenna-75-ohm-fiberglass-surecall-sc-289w-cm-289w

There's an even more powerful antenna available which is "directional". While these catch signals from farther away, it is recommended that they are pointed to nearest cell tower of the carrier for which you want to improve signal. Here's the link to the suggested replacement unidirectional antenna:

https://www.signalbooster.com/products/yagi-antenna-directional-full-band-75-ohm-surecall-sc-231w

Please keep in mind that the zBoost 545 is a 3G-only booster and it will only be a matter of time before the carriers discontinue the 3G service.

If you wish to upgrade the booster kit itself instead of replacing your exterior antenna, here's a 4G equivalent signal booster to what you have but made by a company called weBoost (it happens to own zBoost which it is phasing out to prevent redundancy in its product line). Their basic kit is called the "weBoost Home". Here's the link so you can see this new cell phone booster (3G, 4G, LTE plus possible coverage for upcoming 5G era) and consider purchasing it instead of buying a new external antenna for your 3G booster:

https://www.signalbooster.com/products/wilson-weboost-home-room-472120

What is the characteristic impedance and connector type? Is coax included?

Impedance of CANT0040 antenna is 75 ohm. It has an F Female connector. No coax cable included with purchase, but bracket to mount it is included.