How to work with ManyToMany in Django?

Question

Django

Who can explain on the fingers, and not in the interpreter, how to work with ManyToMany in Django?
Provided
models.py

class Specialization(models.Model):

    spec_name = models.CharField(
        verbose_name="Специализация",
        max_length=150,
        null=False,
        blank=False,
    )


class Person(models.Model):

    author = models.ForeignKey(
        User,
        default=1,
        on_delete=models.PROTECT,
    )
    first_name = models.CharField(
        verbose_name="Имя",
        max_length=150,
        null=False,
        blank=False,
    )
    last_name = models.CharField(
        verbose_name="Фамилия",
        max_length=150,
        null=False,
        blank=False,
    )
    spec = models.ManyToManyField(
        Specialization,
    )

forms.py

class PersonForm(forms.ModelForm):

    class Meta:
        model = Person
        fields = (
            'first_name',
            'last_name',
        )

views.py

def new_person(request):

    if request.method == "POST":
        person_form = PersonForm(request.POST or None)
        if person_form.is_valid():
            instance = person_form.save(commit=False)
            instance.author = request.user
            instance.save()

            messages.success(request, "Создана запись в базе данных!")

            return HttpResponseRedirect('/person/new/')
    else:
        person_form = PersonForm()

    context = {
        'person_form': person_form
    }
    return render(request, 'form.html', context)

My mosk refuses to understand the interpreter.
UPD:
All 3 tables are created in the database. The table of specializations is filled with three records. The page displays a form with the fields: First Name, Last Name, Specializations. Fill it out, hit submit. But the relation in the 'person_specialization' table is not stored in the database. What am I doing wrong? I probably don't keep that same attitude. But how to save it?