User Input only at the end of while loop

You can handle user input that is not a number as shown in https://www.geeksforgeeks.org/python-int-function/.

You should not use break when the user wins or loses. If you use it, the user is not able to play the game again.

To prevent the program from asking every time whether the user wants to play again, you have to check again for remaining tries if attempts == 0 or guess == computerNum.

from random import random
import random

# Guessing Game
attempts = 5

print("===GUESS THE NUMBER===")
print(f"GAME OBJECTIVE: Guess the computer's number in {attempts} tries")

running = True

computerNum = random.randint(1, 15)

while running == True:
    try:
        guess = int(input("Guess the number:\n>"))
    except ValueError as e:
        print("Enter a number")
        continue

    print("\n")

    if guess > computerNum:
        print("Lower...")
        attempts -= 1
        print(f"[You have {attempts} tries left]")

    if guess < computerNum:
        print("Higher...")
        attempts -= 1
        print(f"[You have {attempts} tries left]")

    if attempts == 0:
        print("GAME OVER!!!")
        print(f"The computer's number is {computerNum}")

    if guess == computerNum:
        print("You have guessed the number!!!")
        print("YOU WIN!!!")

    if attempts == 0 or guess == computerNum:
        print("\n")
        playAgain = input("Do you want to continue playing(Y/N):\n")

        if playAgain in ("N", "n"):
            break
        else:
            computerNum = random.randint(1, 15)
            attempts = 5

Like Tacconinja007 said, you could use a try, except statement to make sure the guess is always a number. And if you want to check if the guess is an integer you could do:

if type(guess) == int:

And if you want to make it so the program asks the user if they want to play again once they run out of attempts you could try the following code:

from random import random, randint
import random

# Guessing Game
attempts = 5

print("===GUESS THE NUMBER===")
print(f"GAME OBJECTIVE: Guess the computer's number in {attempts} tries")
running = True

computerNum = random.randint(1, 15)

while running:

try:
    guess = int(input("Guess the number:\n>"))
    print("\n")
except ValueError:
    print("Enter a number")
    continue

if guess > computerNum:
    print("Lower...")
    attempts -= 1
    print(f"[You have {attempts} tries left]")

elif guess < computerNum:
    print("Higher...")
    attempts -= 1
    print(f"[You have {attempts} tries left]")

if attempts == 0:
    print("GAME OVER!!!")
    print(f"The computer's number is {computerNum}")
    replay = input("Do you want to play again? (Y/N): ")
    if replay in ("N", "n"):
        break
    else:
        attempts = 5
        computerNum = random.randint(1, 15)

if guess == computerNum:
    print("You have guessed the number!!!")
    print("YOU WIN!!!")
    break

print("\n")