Ich brauche dazu einen Löffel.

Because brauchen takes a direct object, and direct objects are usually in the accusative case.

• der Löffel = the spoon
• ein Löffel = a spoon in the nominative
• einen Löffel = a spoon in the accusative

So:

• Ein Löffel ist auf dem Tisch. = A spoon is on the table.
• Ich brauche einen Löffel. = I need a spoon.

Since Löffel is masculine, the article changes from ein to einen in the accusative.

Dazu is a very common German word that can mean things like:

• for that
• with that
• for it
• in addition to that

Here it refers back to something already mentioned. So the idea is:

• Ich brauche dazu einen Löffel. = I need a spoon for that.

It is made from:

• da- = referring back to a thing or situation
• zu = to / for / with, depending on context

German often uses these da- compounds instead of phrases like for it or to that.

That is a normal German word order pattern.

German often puts short adverbs like dazu, dann, heute, dort, etc. before a full noun object in the middle of the sentence.

So this sounds natural:

• Ich brauche dazu einen Löffel.

You may also hear:

• Ich brauche einen Löffel dazu.

That can also be correct, but it may sound slightly different depending on context. Very roughly:

• dazu einen Löffel often keeps the focus on what this is for
• einen Löffel dazu can sometimes feel more like a spoon as well / in addition

The sentence you were given is a very natural standard option.

Yes.

The infinitive is brauchen = to need.

Present tense:

• ich brauche
• du brauchst
• er/sie/es braucht
• wir brauchen
• ihr braucht
• sie/Sie brauchen

So Ich brauche simply means I need.

You usually learn German nouns together with their article:

• der Löffel = the spoon

That tells you the gender is masculine.

Once you know that, the accusative form makes sense:

• nominative: ein Löffel
• accusative: einen Löffel

Unfortunately, noun gender in German often has to be memorized. So it is best to learn words as:

• der Löffel
• not just Löffel

Because all nouns are capitalized in German.

So:

• ich brauche = verb, not capitalized
• dazu = adverb, not capitalized
• einen = article, not capitalized
• Löffel = noun, capitalized

This is one of the most noticeable spelling differences between German and English.

Sometimes you could express a similar idea that way, but dazu is much more natural here.

German often prefers words like:

• dafür = for that
• dazu = for that / with that / to that / in addition
• damit = with that / by means of that

Which one sounds best depends on context.

In this sentence, dazu is idiomatic and natural if you mean something like:

• I need a spoon for that
• I need a spoon with that activity / dish / situation

Using für das would usually sound less natural unless you are being very explicit.

Not if you mean one spoon.

In German, a singular countable noun like Löffel normally needs an article or determiner:

• einen Löffel = a spoon
• den Löffel = the spoon
• meinen Löffel = my spoon

So:

• Ich brauche dazu einen Löffel. = correct

Ich brauche dazu Löffel sounds wrong if you mean one spoon.

If you meant plural in a special context, you might say something like:

• Ich brauche dazu Löffel. = I need spoons for that

But even then, speakers would often prefer a determiner such as ein paar Löffel or mehrere Löffel, depending on the situation.

Löffel is roughly pronounced like LUHR-fel, but the ö does not exist exactly in standard English.

A few helpful points:

• ö is a rounded front vowel
• Keep your tongue in a position somewhat like eh or uh, but round your lips

A rough English-friendly guide:

• Löffel ≈ LER-fel or LUHR-fel, but neither is perfect

Also:

• the ff makes the consonant short and crisp
• the final -el is light, almost like -uhl or -el

If you want to sound more natural, listening to native audio is especially helpful for vowels like ö.

Very literally, it is:

• Ich = I
• brauche = need
• dazu = for that / with that
• einen Löffel = a spoon

So the structure is roughly:

• I need for-that a spoon

That sounds odd in English, but it is a useful way to see how German builds the sentence.

Natural English would be:

• I need a spoon for that.

Yes. Benötigen also means to need.

So:

• Ich brauche dazu einen Löffel.
• Ich benötige dazu einen Löffel.

Both are correct.

The difference is mostly style:

• brauchen is the everyday, common word
• benötigen sounds more formal or more written

In normal conversation, brauchen is usually the best choice.