Бабушка сказала, что с костылём я двигаюсь медленнее, зато безопаснее.

Because бабушка is a feminine noun, and in the past tense Russian verbs agree with the subject in gender and number.

• сказал = masculine singular
• сказала = feminine singular
• сказало = neuter singular
• сказали = plural

So Бабушка сказала means Grandma said.

In Russian, что often introduces a subordinate clause, like that in English.

So the sentence is built like this:

• Бабушка сказала = Grandma said
• что с костылём я двигаюсь медленнее, зато безопаснее = that with a crutch I move more slowly, but more safely

Russian normally puts a comma before this kind of что-clause.

Костылём is the instrumental singular of костыль.

After с meaning with, Russian often uses the instrumental case when it means together with / using something.

So:

• костыль = crutch
• с костылём = with a crutch

The ending -ём is a normal instrumental ending for some masculine nouns after a soft stem or spelling pattern.

Both spellings may be seen in informal writing, because Russians sometimes omit ё and write е instead. But the fully correct spelling is костылём.

Using ё helps show both:

• the pronunciation
• the stress

So костылём is pronounced with stress on the last syllable.

Here it means using a crutch or when I have a crutch / with a crutch.

It does not necessarily mean the crutch is just physically next to me. It implies assistance or accompaniment in movement.

Depending on context, a natural English translation could be:

• with a crutch
• using a crutch
• when I use a crutch

Двигаюсь comes from двигаться, which means to move.

It is broader and more general than иду:

• иду = I am going / walking in one direction right now
• двигаюсь = I move / I am moving

In this sentence, the speaker is talking about their overall manner of movement, not just one specific trip somewhere. So двигаюсь fits well.

Yes, it could be omitted in many contexts:

• ...что с костылём двигаюсь медленнее...

Russian often drops subject pronouns when the meaning is clear from the verb ending.

However, я may still be included for:

• clarity
• emphasis
• contrast

Here it helps keep the statement explicit: I move more slowly, but more safely.

They are comparative forms.

• медленнее = more slowly / slower
• безопаснее = more safely / safer

This is something that often surprises English speakers: in Russian, the same comparative form can often correspond to either an English adjective or adverb depending on context.

Here they describe how the person moves, so in English they are best understood as adverbs:

• move more slowly
• move more safely

Russian often prefers the simple comparative form when possible:

• медленнее
• безопаснее

These are shorter and more natural here than:

• более медленно
• более безопасно

The более + adverb/adjective pattern is possible in Russian, but it is usually less compact and often sounds more formal or less idiomatic in everyday speech.

Зато is a very common word that expresses a kind of compensation or trade-off.

The idea is:

• one thing is worse or less desirable,
• but there is a positive side that makes up for it.

So here:

• медленнее = more slowly
• зато безопаснее = but on the other hand, more safely / but at least more safely

A very natural interpretation is:

I move more slowly with a crutch, but it’s safer.

Not exactly.

• но = but
• зато = but, in compensation / but on the plus side / however, the advantage is

Compare the nuance:

• медленнее, но безопаснее = slower, but safer
• медленнее, зато безопаснее = slower, but at least safer / slower, but the good thing is that it’s safer

So зато adds the idea of a positive trade-off.

Russian often omits repeated words when they are easy to understand from context.

So:

• я двигаюсь медленнее, зато безопаснее

literally has only one verb, but the second part is understood as:

• я двигаюсь безопаснее

English does this too sometimes, but Russian is especially comfortable with leaving repeated material out when it is obvious.

Yes, the form itself can correspond to either idea, but the context decides.

Since the verb is двигаюсь (I move), the word describes the manner of movement, so the best English sense is:

• more safely

If it described a noun, then it could be safer as an adjective.

For example:

• Этот способ безопаснее. = This method is safer.

A learner might want to notice these stress patterns:

• ба́бушка
• сказа́ла
• костылём
• дви́гаюсь
• ме́дленнее
• безопа́снее

Stress matters in Russian, and in forms like костылём it is especially useful because the written ё also points you to the stressed syllable.