Summary of Key Points
What is a muscle pump?
A “muscle pump” describes a state of cellular swelling in muscle tissue caused by hyperhydration (excess fluid accumulation in muscle cells).
How do muscle pumps contribute to hypertrophy?
When a muscle cell swells, osmosensors in the cell wall detect it as a threat to the cell's integrity and it reacts by reinforcing its structure.
How can I maximize the Muscle Pump Effect?
During Training, constant tension is key; eliminate pauses between reps, reduce your rest intervals, and modify your range of motion to maintain constant tension. Nutritional factors such as hydration also come into play.
What is a Muscle Pump?
A “muscle pump,” known by scientists as “weightlifting-induced cellular swelling,” is a sensation of tightness, swelling, and mild burning caused by high-volume weightlifting (but if you’ve spent any time around weights, you already knew that). It is a form of metabolic stress, one of the three mechanisms of muscular hypertrophy pursuant to resistance training. Physiologically, a muscle pump is the result of fluid accumulation in muscle tissue during periods of restricted blood flow.
Let's break that down: in the cardiovascular system, oxygenated blood flows from the heart towards your body’s cells via tiny blood vessels called capillaries. Capillary walls are thin enough to allow fluids, oxygen and other nutrients to diffuse out and be absorbed by surrounding cells.
During periods of constant & intense muscle contractions, the veins returning blood to the heart are compressed, causing blood to accumulate in the area. When the buildup of blood becomes sufficient, blood plasma - the watery substance that carries red blood cells - begins to seep out of the vessels and into muscle cells by way of osmosis, creating the cellular swelling effect. 
Now that we’ve sorted that out, let’s move on to this article’s main concern...
How does "Getting a Pump" Build Muscle?
Actually, cellular swelling doesn't just build muscle; it regulates many cellular processes. [2-7] Here are some examples:
Increases amino acid uptake  
Increases glycogen synthesis [11-15]
Muscle growth occurs when the net balance of protein synthesis and breakdown is positive  - and this is exactly the environment that the muscle pump effect creates.
Why? The leading theory holds that increased pressure within muscle cells caused by high-volume weightlifting is percieved as a threat to cell structure. The cell reacts by taking up extra nutrients, replenishing its fuel reserves, and reinforcing its structure.  
“You should see the muscle pump as a tool to augment your training, not your primary focus”
Keep in mind that cellular swelling is not the sole driver of muscle growth, and it should not necessarily be the focus of your training. Most researchers regard mechanical overload - strength gains, essentially - to be the most important driver of muscle growth, followed by metabolic stress. You should see the muscle pump as a tool to augment your training, not your primary focus.
The Psychological Aspect
Compared to traditional strength training, the type of training which produces a muscle pump (i.e. high volume, moderate load, and low rest periods) also produces a greater β-endorphin response. This might explain the coveted status of the muscle pump effect. People like endorphins.
It's also worth mentioning that the muscle pump likely has other psychological effects. The acute boost in muscle size and vascularity would almost certainly enhance confidence and motivate training performance.
Techniques to Maximize the Muscle Pump Effect
In the Gym
Maintain Tension During Sets
Remember that the muscle pump effect is caused by contracting muscles restricting blood flow. When these muscles relax, blood flow normalizes and the muscle pump effect is reduced.
You can maintain constant tension by modifying your lifting tempo & range of motion. Eliminating pauses between reps in particular has been shown to increase muscle strength and size.  
Reduce Rest Intervals
The longer your rest intervals, the more time your muscle cells have to restore fluid balance to baseline. For the sections of your workout that you want to emphasize metabolic stress, reduce rest times.
Note: shorter rest times aren't always better. Reducing rest times sacrifices strength performance in exchange for the added metabolic stress, so be strategic about how you program this technique.
Balance your Program
Metabolic stress training should be programmed separately from strength training so that they do not interfere. Generally, strength training should come first in a workout, with metabolic stress coming later or in a different session altogether.
If you want to learn more about programming, check out our free email course on training for muscular hypertrophy.
Nutrition & Supplementation
Just like with weight training in general, the training techniques above work best when the bodily is properly nourished. Here arethe most significant nutritional factors for inducing a muscle pump:
It’s called hyperhydrative cellular swelling for a reason; in a dehydrated state, you’ll have less fluid available for muscle cell expansion.