When it comes to bodybuilding, the process of muscle growth can be different depending upon the training program that is utilized.
There are two main ways that muscles will grow: sarcoplasmic hypertrophy and myofibril hypertrophy. How you train will determine which type of growth you are primarily experiencing.
Sarcoplasmic hypertrophy is what is going to mostly make the actual size of your muscle look bigger initially. You've likely experienced this before if you ever put in a really good bicep and tricep workout and noticed your arms looked particularly large. What's going on?
What happened is that there was a build-up of intercellular byproducts (such as lactic acid for example) that caused a so-called 'muscle pump'.
Unfortunately, as you've also probably noticed before, this 'pump' does not last for very long. Within an hour or two after the workout, those byproducts will be removed from the tissue and you'll be back looking more like your usual size.
Not that obviously this type of growth isn't beneficial - a good muscle pump can be highly motivating and keep you coming back to the gym for more, it's just important to understand that this is not going to be long-term growth that's going to stick with you if you stopped working out for a week.
Next we come to the other type of hypertrophy - the type that's more 'true muscle growth' because it tends to last for months after it's experienced. Myofibril hypertrophy occurs when the actual muscle tissue (also known as the 'myofibril', hence the name) enlarges in response to an overloading stimulus.
These types of gains are a lot slower to be seen and usually won't produce quite the same amount of size as the sarcoplasmic hypertrophy does. But, the good news is that these gains are also associated with a good increase in strength, thus it's a great indication that you are improving in not only size but also your performance in the gym.
Now, with most training programs you are going to get the best of both worlds just because of the nature of training. First you'll see the sarcoplasmic growth, and then if you carry on with your workout and progressively keep adding weight to the bar then you're going to see the myofibril hypertrophy.