To get a mech in Biomutant you will first need to speak to an NPC called Gizzmo in the chuck station. Gizzmo will give you a Mech as a part of the Main quest for the Dark lands area of the game. However, before you can get the mech you will need to first capture six creatures for Gizzmo and gather a few mech parts - he will also tell you where you can find them all.
But in short, you will need to get past a few Hypoxia zones that will inflict damage on you the longer you are in them. Because of this, you will need to be quick, but you shouldn't have too much trouble getting past them. Just be quick and quickly find a door marked by a quest marker, and follow the other markers.
Related: All Psi-Powers in Biomutant
Past the door, you can find the part you need, and after you grab it, you will need to bring it back to Gizzmo to complete the quest. That said, after you complete this quest, you will be able to use the mech again by visiting a Mekton docking station. You can find these locations on your map via yellow dots, and to get the mech from them, you will need to press down on the D-pad, or press 4 on your keyboard.
Once you have done this, you will summon your brand new mech and then use it to fight enemies or transverse terrian like other mounts.