In the upper body, when swimming front crawl, you’ll use the deltoids, latissimus dorsi (down the side of your back), trapezius, triceps and biceps muscles.  The muscles of the shoulders and around shoulder blade (including the deltoids) will help ‘hold’ the ‘paddle’ [your arm] in place as your body moves past it.  Because the shoulder muscles, also known as the rotator cuff, work so hard in front crawl, they can become damaged leading to one of the most common injuries swimmers can get is swimmer's shoulder aka subacromial impingement.

Your core muscles, including your trapezius and latissimus dorsi, help you hold a streamlined torso in front crawl 

How to engage your core to boost swim speed

In the lower body, front crawl works the hip flexors, quads, hamstrings, glutes and calves. Your abdominal muscles will also tone up from stabilising you in the water.

Rotator cuff: What it is, how it works and what happens if it becomes damaged

Is there a difference between front crawl and freestyle?

Strength training for swimming: 6 key gym exercises

A schematic showing some of the skeletal muscles of a Homo sapiens. The schematic was based on an image in the book "The human body" by Linda Gamlin

(Licensed under Wikimedia Commons https://commons.wikimedia.org/wiki/File:Skeletal_muscles_homo_sapiens.JPG)

Muscle activation exercises: how they work and why they are important

How to relieve muscle soreness after a hard workout

Do muscles have memories?

What's the difference between muscles, tendons and ligaments?

What is a physiotherapist and what do they do?

What happens to your body if you stop training?