Новая гуашь ярче старой, поэтому рисунок получился красивее.

Because гуашь is a feminine noun in Russian.

So the adjective has to agree with it:

• новая гуашь = new gouache/paint
• старой (гуаши) = of the old gouache / than the old gouache

Even though гуашь ends in a soft sign (-ь), it is feminine. That is something learners often just have to memorize, because nouns ending in -ь can be either masculine or feminine.

Ярче is the simple comparative form of яркий (bright), so it means brighter.

Russian often prefers these short comparative forms:

• яркий → ярче = brighter
• красивый → красивее = more beautiful

You can sometimes use более + adjective too, but with common adjectives like these, the simple comparative is more natural.

So:

• Новая гуашь ярче старой = The new gouache is brighter than the old one

Because after a simple comparative like ярче, Russian often uses the genitive case for the thing being compared.

So:

• ярче старой literally means brighter than the old one
• старой is genitive singular feminine

The full underlying phrase would be:

• ярче старой гуаши

But гуаши is omitted because it is obvious from context.

Russian can also use чем for comparisons:

• Новая гуашь ярче, чем старая

That also means The new gouache is brighter than the old one.

Yes. That is exactly what is happening.

Russian often leaves out a noun if it is clear from context. Here, старой stands in for старой гуаши.

This is similar to English the old one, where one replaces the noun.

So:

• новая гуашь = the new gouache
• старой = the old one / the old gouache

Поэтому means therefore, so, or that’s why.

It introduces the result of the first part:

• Новая гуашь ярче старой = the cause/reason
• поэтому рисунок получился красивее = the result

The comma is used because Russian is joining two clauses here.

So the sentence structure is basically:

• X, поэтому Y
• X, so Y

Literally, it is something like:

• the drawing turned out more beautiful

The verb получиться often means:

• to turn out
• to come out
• to end up being

So рисунок получился красивее means the final result was better-looking.

This is a very natural Russian way to talk about results, especially with creative work, food, plans, photos, and so on:

• Торт получился вкусным = The cake turned out tasty
• Фото получилось удачным = The photo turned out well

Because the verb here is получиться, not получить.

These are different verbs:

• получить = to receive / to get
• получиться = to turn out / to come out

So:

• рисунок получился красивее = the drawing turned out more beautiful
• not the drawing received...

Also, получился is masculine singular because рисунок is masculine singular.

Because the sentence is using a comparative, not a plain adjective.

• красивый = beautiful
• красивее = more beautiful

The idea is not just the drawing was beautiful, but the drawing turned out more beautiful.

So the comparative is needed.

Also, after получиться, Russian can use a comparative naturally:

• получился лучше = turned out better
• получился красивее = turned out more beautiful

It is implied by context.

Красивее means more beautiful, but Russian does not always need to spell out the second item if it is obvious. Here the idea is something like:

• more beautiful than before
• more beautiful than it would have been with the old gouache
• simply better-looking as a result

English does this too sometimes:

• It came out better.
• Better than what? Context tells you.

So the sentence is perfectly natural even without an explicit than... after красивее.

Because Russian has no articles.

So гуашь can mean:

• gouache
• the gouache
• some gouache

And рисунок can mean:

• a drawing
• the drawing

Context tells you which meaning is intended. In this sentence, English usually translates them as the new gouache, the old one, and the drawing, because the situation sounds specific.

It is understandable, but it sounds less natural.

Russian usually prefers:

• ярче rather than более яркая
• красивее rather than более красивым

Also, after получился, the comparative красивее is especially natural.

So the original sentence sounds better:

• Новая гуашь ярче старой, поэтому рисунок получился красивее.

They are:

• новый = new
• гуашь = gouache, poster paint
• яркий = bright
• старый = old
• поэтому = therefore, so
• рисунок = drawing, picture
• получиться = to turn out, to come out
• красивый = beautiful

This can help you see how the forms in the sentence were built:

• новый → новая
• яркий → ярче
• старый → старой
• получиться → получился
• красивый → красивее

A helpful stress guide is:

• Но́вая гуа́шь я́рче ста́рой, поэ́тому рису́нок получи́лся краси́вее.

A few pronunciation notes:

• ярче sounds roughly like YAR-che
• старой has stress on the first syllable: STA-roy
• поэтому is stressed on э́
• получился is stressed on чи́
• красивее is stressed on си́

For many learners, красивее is tricky: it is кра-си́-ве-е, not stressed at the end.