Me gusta lo bien que mi profesora explica la gramática.

Because gustar works differently from to like in English.

In English, I like X makes I the subject.
In Spanish, gustar is more like X is pleasing to me.

So in this sentence:

• me = to me
• gusta = is pleasing
• lo bien que mi profesora explica la gramática = the thing that is pleasing to me

So Me gusta... literally works like I like..., but structurally it is closer to ...pleases me.

Because the subject is treated as one whole idea:
lo bien que mi profesora explica la gramática = how well my teacher explains grammar

That whole clause is singular, so Spanish uses gusta.

Compare:

• Me gusta el libro. = I like the book.
• Me gustan los libros. = I like books.
• Me gusta cómo canta. = I like how she sings.

A whole clause or action is normally singular, so gusta is correct.

Lo bien que is a very common structure meaning how well or the way ... so well.

In this sentence:

• lo bien que mi profesora explica la gramática

means something like:

• how well my teacher explains grammar
• the fact that my teacher explains grammar so well

This pattern is:

• lo + adjective/adverb + que

Examples:

• Me gusta lo rápido que corre. = I like how fast he runs.
• Me sorprendió lo fácil que fue. = I was surprised by how easy it was.
• No sabes lo bien que canta. = You don’t know how well she sings.

So lo bien que is not just well; it adds emphasis.

Because que introduces the clause that explains what is being described.

So:

• lo bien by itself = how well / how good
• lo bien que mi profesora explica la gramática = how well my teacher explains grammar

That que links the idea how well to the rest of the sentence.

It is part of the fixed structure lo + adjective/adverb + que.

Yes, absolutely.

Me gusta cómo mi profesora explica la gramática is natural and correct. It means I like how my teacher explains grammar.

The difference is mainly one of emphasis:

• Me gusta cómo mi profesora explica la gramática. = neutral, straightforward
• Me gusta lo bien que mi profesora explica la gramática. = more emphatic, stressing how well she explains it

So both are good, but lo bien que highlights the quality of her explanation more strongly.

No. Me gusta... is already complete.

The me already means to me, so a mí is optional. You add a mí mainly for emphasis, contrast, or clarity.

Examples:

• Me gusta lo bien que mi profesora explica la gramática. = normal
• A mí me gusta lo bien que mi profesora explica la gramática. = I like how well my teacher explains grammar

That second version might be used if you are contrasting your opinion with someone else’s:

• A Juan no le gusta, pero a mí me gusta...

Mi profesora means my female teacher.

A few points:

• mi = my
• profesora is the feminine form
• if the teacher were male, it would be mi profesor

Also, Spanish usually does not use an article before a possessive in this kind of structure. So you say:

• mi profesora
• not la mi profesora

That old-style form exists only in very limited or archaic uses, not in normal modern Spanish.

Because gramática is a noun, and in Spanish it is very common to use the definite article with school subjects, abstract nouns, and general categories.

So:

• la gramática = grammar

This sounds natural in Spanish, even where English might simply say grammar without the.

Compare:

• Estudio la gramática española. = I study Spanish grammar.
• La historia me interesa. = History interests me.
• Le gusta la música. = He/She likes music.

So explica la gramática is the normal way to say explains grammar.

Explica is the present tense of explicar: she explains.

Here it usually suggests a habitual or general idea, not necessarily something happening right this second.

So the sentence means something like:

• I like how well my teacher explains grammar.
• I like the way my teacher explains grammar.

It suggests this is her usual way of explaining, or a general quality she has.

Sometimes, but the meaning changes slightly.

• explicar = to explain
• enseñar = to teach

So:

• Me gusta lo bien que mi profesora explica la gramática.
  = I like how well my teacher explains grammar.

• Me gusta lo bien que mi profesora enseña gramática.
  = I like how well my teacher teaches grammar.

Both can work, but explica focuses on the clarity of her explanations, while enseña is broader and refers to her teaching in general.

Because gramática is stressed on the third-to-last syllable:

• gra-MÁ-ti-ca

Words stressed on the third-to-last syllable are called esdrújulas, and in Spanish they always take a written accent.

So the accent in gramática is required.