How to get an input from user in Pygame and save it as a variable?

There's nothing baked into Pygame for this. You will either need to use a 3rd-party GUI library, or build it yourself. Example: if the textbox has focus, take all keydown events, and append to a string. Each frame, draw a box, then draw the string on top.

Building a simple one shouldn't be that hard, but if you want a more full-featured one, it will likely be easier to use a library.

I am currently assuming this function worked with your program is successful and raises no errors. Here is a function in the link you gave us:

def ask(screen, question):
    "ask(screen, question) -> answer"
    pygame.font.init()
    current_string = []
    display_box(screen, question + ": " + string.join(current_string,""))
    while 1:
      inkey = get_key()
      if inkey == K_BACKSPACE:
        current_string = current_string[0:-1]
      elif inkey == K_RETURN:
        break
      elif inkey == K_MINUS:
        current_string.append("_")
      elif inkey <= 127:
        current_string.append(chr(inkey))
      display_box(screen, question + ": " + string.join(current_string,""))
    return string.join(current_string,"")

It looks like this is how you get input from a user with a pygame screen right? Let's look at line 4 of this function:

current_string = []

The stuff the user types in is stored in this list. Assuming you know how to take a string and put it on the screen, you could save the string like this:

string_input = ''.join(current_string)

If you can make a similar function (if this one doesn't work), you can do the same thing! Just save the first item in the list holding the string in a variable as shown above. If you have any problems, please comment so I can edit my answer.
To the next part now. You can simply activate this function at any time. You might want to activate when something happens. An example is when the snake eats the apple. You probaly have a function for that I believe. You can make a variable like this :

Eat = 0

And put in that function. When that variable is equal to 0. Nothing really activate the other function. When the snake eats the apple, reset the variable to 1 and then activate the function like this:

if Eat = 0:
    someclassname.ask()

You do this with many other occasions. I hope this is clearer and more helpful!

Have you tried getting key event's and then putting them together to form something? You could put something like this inside of your main game loop.

input = ''
for event in pygame.event.get():
        if event.type == pygame.KEYDOWN:
            if event.key == pygame.K_ESCAPE:
                playing = False
            if event.key == pygame.K_w:
                input = input + "w"
            if event.key == pygame.K_s:
                input = input + "s"
            if event.key == pygame.K_a:
                input = input + "a"

Then you can check for when the user is done with input(button/enter key), and finalize the variable. With this you might run into a problem where the key is held longer so you get 3 w's for only pressing the button once. If this is the case you can either 1. not allow another input until a certain time has passed(.25 s maybe?) or 2. use pygame.KEYUP instead of pygame.KEYDOWN and exclusively check for strokes.