how assemblers run interrupts

The code you posted is real mode 16 bit assembler. Those days, it was perfectly okay to invoke interrupts directly, and you could even do it from languages like C, Pascal and Basic (depending on the compiler you used).

Btw, the assemblers themselves do not invoke any interrupts, they just generate the required machine code for calling an interrupt. At run time, it's the CPU that executes the code and when it encounters an interrupt instruction, the interrupt is fired.

In modern day protected mode development, you can still call interrupts (in a fashion), but you need to be running in kernel mode. If you are serious about learning assembly language, I would advise you to get a more updated book. Don't waste too much time studying 16 bit assembly, you might as well go learn the PDP-11 instruction set in that case.

They cannot.
Windows, for instance, won't allow such a call (please, Google for 'protected mode').
:)

INT 10H isn't an interrupt, strictly speaking.

An interrupt is when a piece of hardware needs to have something done, and raises a special kind of alert to the processor to get it's attention immediately. It does this so that action can be taken as quickly as possible, rather than waiting until the processor has nothing else to do.

For example, a physical serial port may receive a character, and it needs to hand it off to the processor before another one is received and it has no-where to put it. (In practice they are buffered, but the interrupt still happens).

The processor gets the alert signal, saves what it is doing, works out which hardware signalled the interrupt, and begins executing the code to handle that particular interrupt. When it is finished, it resumes exactly where it left off.

INT 10H and it's other little pals do not do that: they are software calls to OS methods, and are a way to elevate a user so that he can do privileged operations, that he is normally forbidden to do to prevent him messing things up.