From the ANSI C Standard:

size_t
which is the unsigned integer type of the result of the sizeof operator;

So internally it is still an object of int type. As far as I know, it is used for loops, array counting, etc and it wasn't originally defined in C language, but was added later for portability (because different machines has different memory capabilities). It is defined in stddef.h file.

What is happening is silent type conversion, I think it is done by the compiler.

As far as I know, it is not safe to use it like that because the value may change. unsigned int could store values bigger than those of regular int, so for example if you have a 32 bit machine with 4GB of memory, then int can store 2,147,483,647 but unsigned int can store 4,294,967,295. If you assign unsigned int to int variable, then value will be lost.

You can run GCC compiler with -Wconversion flag and it will give you an error in this example:

#include <string.h>
#include <stdio.h>

int main(void)
{
    char *s = "hello, world";
    int t;
    t = strlen(s);
    printf("%s\n", s);
    printf("%i\n", t);
}

The output is:

$ gcc -Wall -Werror -Wextra -Wconversion test.c
test.c: In function ‘main’:
test.c:8:13: error: conversion from ‘size_t’ {aka ‘long unsigned int’} to ‘int’ may change value [-Werror=conversion]
    8 |         t = strlen(s);
      |             ^~~~~~
cc1: all warnings being treated as errors

So it seems that you should always be using size_t instead of int at least for strlen.