Learn Russian: Мой брат думает, что мотоцикл менее надёжный, чем машина. - My brother thinks that a motorcycle is less reliable than a car.

Because we are using чем (than) to compare two subjects. When you use чем for comparison, the thing you are comparing to (машина) takes the same case as the first thing (мотоцикл). Since мотоцикл is the subject of its clause (nominative), машина is too.

How does the phrase менее надёжный (less reliable) work?

To say 'less [adjective]' in Russian, you simply use the word менее (less) followed by the normal adjective. While 'more' often has a special single-word comparative form, 'less' almost always uses this two-word structure.

Why is the adjective надёжный in the masculine form if we are also talking about a машина (a feminine word)?

The adjective надёжный describes the мотоцикл (motorcycle), which is a masculine noun. The sentence is saying the motorcycle is the 'less reliable' thing, so the adjective must match the motorcycle's gender.

In English, I could say 'My brother thinks a motorcycle is less reliable', dropping the word 'that'. Can I drop что in Russian?

No, you cannot drop it. In Russian, the conjunction что is strictly required to connect the first part of the sentence (Мой брат думает) to the second part.