Cet imperméable résiste mieux à la pluie que mon vieux gilet.

Because imperméable starts with a vowel sound.

In French, the masculine singular demonstrative changes like this:

• ce
  • consonant sound: ce gilet
• cet
  • vowel or mute h: cet imperméable
• cette
  • feminine noun: cette veste

So cet is just the form used to make pronunciation smoother before a vowel sound.

Here it is a noun: un imperméable = a raincoat / waterproof coat.

But imperméable can also be an adjective, meaning waterproof:

• un manteau imperméable = a waterproof coat
• ce tissu est imperméable = this fabric is waterproof

So the same word can be either an adjective or a noun depending on how it is used.

Because résister is normally followed by à in French.

So you say:

• résister à la pluie
• résister au froid
• résister à la pression

This is just the standard construction of the verb. English learners often want to translate directly from resist and forget the preposition, but in French résister usually needs à before what is being resisted.

Because mieux is the comparative form of bien, and here it modifies the verb résiste.

Compare:

• Il chante bien. → Il chante mieux.
• Cet imperméable résiste bien à la pluie. → Cet imperméable résiste mieux à la pluie.

Use meilleur when you are describing a noun:

• un meilleur manteau
• une meilleure solution

So:

• résiste mieux = resists better
• un meilleur imperméable = a better raincoat

Because adverbs like bien, mieux, mal, vite, etc. often come after the conjugated verb in French.

So:

• Il parle bien.
• Elle court vite.
• Cet imperméable résiste mieux à la pluie.

That word order is normal in French.

Here que means than in a comparison.

The pattern is:

• plus ... que = more ... than
• moins ... que = less ... than
• mieux ... que = better ... than

So:

• Cet imperméable résiste mieux à la pluie que mon vieux gilet.

means that the raincoat performs better in resisting rain than the old cardigan/vest.

Because French comparisons often leave out repeated material when it is understood.

The full idea is something like:

• Cet imperméable résiste mieux à la pluie que mon vieux gilet ne résiste à la pluie.

But that would be unnecessarily heavy. French, like English, usually shortens it:

• ... mieux à la pluie que mon vieux gilet

The missing part is understood from the context.

Because French normally uses an article with nouns in general statements like this.

So la pluie means the rain, but in many contexts French uses the where English would simply say rain:

• à la pluie = literally to the rain, but naturally to rain
• au soleil = in the sun / to sunlight
• au froid = to the cold

This is very common and natural in French.

Because vieux is one of the common adjectives that often come before the noun.

Many French adjectives come after the noun, but some very common ones often come before it, especially short everyday adjectives like those in the usual BAGS group:

• Beauty
• Age
• Goodness
• Size

Vieux relates to age, so it commonly goes before the noun:

• un vieux gilet
• une vieille maison
• un jeune homme

Not necessarily. Gilet can refer to different kinds of sleeveless or knitted garments depending on context.

Common possibilities include:

• cardigan
• vest
• waistcoat

So the exact English translation depends on context, region, and the image in the learner’s materials. Grammatically, though, mon vieux gilet is simply my old gilet/vest/cardigan.

Because mon vieux gilet is a normal possessive noun phrase meaning my old gilet.

French often uses possessive adjectives just like English:

• mon = my masculine singular noun
• ma = my feminine singular noun
• mes = my plural nouns

Since gilet is masculine singular, mon is correct.

You could say le mien in another version, but that would change the structure:

• Cet imperméable résiste mieux à la pluie que le mien.

That would mean than mine, with the noun left unstated.

Yes, the words flow together quite smoothly.

A rough pronunciation is:

• cet imperméable ≈ set-tan-pair-may-abl

A few useful points:

• cet sounds like set
• the final t of cet is pronounced because the word itself ends in t
• imperméable has several syllables: im-per-mé-a-ble
• in natural speech, the transition between cet and imperméable is smooth because the second word starts with a vowel sound

You do not need to over-separate the words.

Yes. A very natural alternative would be:

• Cet imperméable est plus résistant à la pluie que mon vieux gilet.

That uses the adjective résistant instead of the verb résister.

Both are correct, but they are built differently:

• résister à = a verb structure
• être résistant à = an adjective structure

The original sentence is perfectly natural and a bit more dynamic because it focuses on the action or performance of the raincoat.