Ég held ekki fast í reipið þegar það er blautt.

Halda is the infinitive, meaning to hold. After ég you need the present-tense form for I, which is held.

So:

að halda = to hold
ég held = I hold

This verb is irregular, so the vowel changes from a to e.

Because Icelandic main clauses usually follow the verb-second rule.

In this sentence:

Ég = first element
held = finite verb, so it comes second
ekki = comes after that

So Ég held ekki ... is the normal order.

If another element came first, the verb would still stay in second position, for example:

Þegar það er blautt held ég ekki fast í reipið.

Here fast means firmly or tightly. It is not the English adjective fast meaning quick.

In halda fast í, it describes how you are holding the rope: with a firm grip.

So:

halda í reipið = hold onto the rope
halda fast í reipið = hold tightly onto the rope

Because halda í is a common Icelandic expression meaning hold on to something.

The preposition is part of the verb phrase here, not just a random extra word.

So:

halda í e-ð = hold on to something

This is different from some other uses of halda, which can mean things like keep, maintain, or think depending on the construction.

It is the definite article: the.

The basic noun is reipi = rope.
With the definite article attached, it becomes reipið = the rope.

Icelandic usually puts the at the end of the noun instead of using a separate word.

With halda í, the object is normally in the accusative.

So here reipið is accusative. The reason it looks the same as the basic form is that reipi is a neuter noun, and in neuter singular the nominative and accusative are often identical.

So even though the case is doing grammatical work, you do not see a visible change here.

Það means it, and it refers back to reipið.

Since reipi is a neuter noun, the pronoun referring to it must also be neuter singular:

reipið → það

So það er blautt means it is wet, where it = the rope.

Because adjectives in Icelandic agree with the noun or pronoun they describe.

Here blautt describes það, which refers to reipið, a neuter singular noun. So the adjective must also be neuter singular:

blautur = masculine
blaut = feminine
blautt = neuter

That is why the sentence has það er blautt.

Because þegar introduces a subordinate clause, and subordinate clauses do not use the same verb-second pattern as main clauses.

So the normal order is:

þegar + subject + verb

Hence:

þegar það er blautt

That is the expected word order after þegar.

Yes, you could say:

Ég held ekki í reipið þegar það er blautt.

That still makes sense, but it is slightly less specific. Without fast, you are just saying you do not hold onto the rope. With fast, you add the idea of firmly/tightly.

So fast adds nuance rather than being absolutely required.

They are related, but not the same.

halda í usually means hold on to or grip something.
halda á usually means hold/carry in your hand.

So in this sentence, halda fast í reipið suggests gripping the rope securely, which fits the context very well.

It can very naturally express a general rule or habit, just like the English simple present.

So the sentence can mean something like:

I don’t hold onto the rope tightly when it’s wet
as a general statement, not only at this exact moment.

That use of the present tense is completely normal in Icelandic.