Góð beygingartafla fær mig til að skilja þágufall og eignarfall betur.

Because góð beygingartafla is the subject of the sentence, so it is in the nominative singular.

• beygingartafla is a feminine singular noun.
• The adjective góður has to agree with it in gender, number, and case.
• So the correct form here is góð = feminine, singular, nominative.

You would expect góða in some other environments, for example with certain object forms or with weak adjective usage, but not here.

The verb fær is in the 3rd person singular, and góð beygingartafla is the noun phrase that matches it as the thing doing the action.

So the basic structure is:

[Góð beygingartafla] [fær] [mig] [til að skilja ... betur].

That means:

• góð beygingartafla = the thing causing something
• fær = gets / makes
• mig = me
• til að skilja ... betur = to understand ... better

So the table is what gets me to understand.

Yes, fá often means get or receive, but in this sentence it is being used in a very common construction:

fá einhvern til að gera eitthvað

This means:

to get someone to do something
or
to make someone do something

So here:

Góð beygingartafla fær mig til að skilja ... betur
= A good declension table gets me to understand ... better
or more naturally, helps me understand ... better.

Also, fær is the present tense singular form of fá.

Because mig is the accusative form of ég.

In the pattern fá einhvern til að ..., the person being influenced is treated as the object of fá, so Icelandic uses the accusative:

• ég = I
• mig = me
• mér = to me / for me

So:

• fær mig til að skilja = gets me to understand

Not:

• fær ég — incorrect here
• fær mér — also incorrect here

til að introduces an infinitive clause, and in this sentence it works much like English to in to understand.

In the expression:

fá einhvern til að gera eitthvað

the til að + infinitive part tells you what someone is caused or persuaded to do.

So:

• fær mig til að skilja = gets me to understand

This is a very common Icelandic pattern.

Because it comes after til að.

After til að, Icelandic normally uses the infinitive form of the verb:

• til að skilja = to understand
• til að læra = to learn
• til að lesa = to read

So skilja is not conjugated here because it is not the main finite verb of the sentence. The finite verb is fær.

They are the things being understood, so they function as the object of skilja.

The verb skilja normally takes an accusative object. That means these nouns are effectively in the accusative here.

However, both þágufall and eignarfall are neuter singular nouns, and for many neuter nouns in Icelandic, the nominative and accusative forms are the same. So the form does not visibly change.

That is why you see:

• skilja þágufall
• skilja eignarfall

rather than some obviously different-looking accusative form.

Because the sentence is talking about these cases in general, as grammar concepts, not about a specific individual thing.

In Icelandic, when you talk about abstract ideas, school subjects, grammatical categories, languages, and similar concepts, you often do not use the definite article.

So here:

• skilja þágufall og eignarfall betur
  means understand the dative and genitive better in a general sense.

That is more natural than making them definite in this context.

Because betur is an adverb, while betra is usually an adjective form.

Here, the word is modifying the verb skilja:

• skilja betur = understand better

Since it describes how well someone understands, Icelandic uses the adverb betur, which is the comparative of vel (well).

Compare:

• Ég skil þetta vel. = I understand this well.
• Ég skil þetta betur. = I understand this better.

But betra would be used when describing a noun or pronoun as better in an adjective sense.

Yes. Very much so. The most useful pattern here is:

X fær Y til að + infinitive

meaning:

X gets Y to do something
or
X makes Y do something
or sometimes more naturally
X helps Y do something

Your sentence follows that pattern exactly:

Góð beygingartafla fær mig til að skilja þágufall og eignarfall betur.

You can build many similar sentences from it, for example:

• Kennarinn fær mig til að æfa meira. = The teacher gets me to practice more.
• Þessi bók fær mig til að hugsa. = This book makes me think.
• Góð skýring fær nemendur til að skilja betur. = A good explanation helps students understand better.

So this sentence is useful not just for vocabulary, but as a model for a very practical grammar structure.