ring-flip
A cyclohexane chair is not locked in place. The ring can wriggle from one chair into a second, equally valid chair without breaking a single bond, by flipping its puckers — the carbon that was tipped up swings down and the one that was down swings up. This conformational somersault is the ring-flip, and a cyclohexane is doing it constantly at room temperature.
The crucial consequence is bookkeeping: a ring-flip turns every axial position into an equatorial one and every equatorial into axial. A hydrogen (or substituent) that was pointing straight up as an axial bond ends up tilted out as an equatorial bond after the flip, and vice versa — yet it stays on the same face of the ring (an 'up' group stays up). The flip swaps axial and equatorial, but it does not swap which side of the ring a group is on.
Because the flip is fast, a substituted ring is really a moving equilibrium between two chairs, and the molecule spends more time in whichever chair is lower in energy. For a single bulky group, that is the chair with the group equatorial, so the equatorial-rich chair dominates. The ring-flip is the engine of all this: it is the mechanism by which a six-membered ring 'shops around' for its most comfortable shape, and reasoning about it is how you predict which conformer a real molecule prefers.
Flip the chair of methylcyclohexane and the methyl moves from equatorial (roomy) to axial (crowded); the molecule prefers the first chair, so it spends most of its time with the methyl equatorial.
A ring-flip swaps axial and equatorial, but a group keeps its up/down face.
A ring-flip does not change which face (up or down) a group is on, so it cannot turn cis into trans — that would require breaking bonds. It only trades axial for equatorial.