What does it mean to say that mass "approaches infinity"? What does it mean to say that mass "approaches infinity"?
I have read that mass of a body increases with the speed and when the body reaches the speed of light, the mass becomes infinity. 
What exactly does it mean to say that the mass "approaches infinity" or "becomes infinity"? I am not able to get a picture of "infinite mass" in my mind.
 A: It means that its inertia (resistance to change in the state of motion) approaches infinity. You probably already know from $F=ma$ that for the same change in speed (acceleration), a larger mass requires a larger force. As the velocity of a body approaches the speed of light, its inertia (i.e. $\gamma m$) becomes so high that you need an impossibly infinite force to accelerate it to exactly the speed of light.
And as Georg says, whenever you see "becomes infinite", read it as "approaches infinity".
A: As the speed of a body approaches the speed of light, it becomes harder and harder to accelerate body. This fact is captured by following formula.
$$f(v) = \frac{m_0}{\sqrt{1-\frac{v^2}{c^2}}}$$
When a physicist says that, 
when speed of a body equals the speed of light its mass becomes infinite.
He is not saying $f(v = c)$ but $\lim_{v\to c}f(v) = \infty$
You might want to read  about limit.
A: The answers given so far are fine, but to my surprise nobody's mentioned the most important point: in modern terminology, we generally don't say that the mass of an object increases with speed. "Relativistic mass increase" is outdated terminology, not used by most physicists anymore. In general, nowadays, "mass" means "rest mass" and is independent of velocity. Igor Ivanov's answer to this question says it all. 
I haven't read the article by Lev Okun that he refers to, but I like the term "pedagogical virus" for this notion.
A: It means that if a finite force acts on the mass in its stationary frame, then in all other frames as the measured velocity approaches c, the acceleration approaches 0 which implies the "relativistic mass", the "apparent mass" the "effective mass", call it what you will, approaches infinity.
A: The mass will never be infinite because it would require infinite work to reach required speed. It's just a figure of speech.
Instead of trying to imagine "infinite mass" try imagining the process: as you apply force ("press the accelerator pedal") the object instead of going faster, gets heavier. The more work you provide, the heavier the object gets.
A: If you want to imagine an infinite mass just think of a piece of matter which does not response (i.e. accelerate) to any force. Obviously, it is an ideal concept. In reality "the mass approaching infinity" means it is increasing without any finite upper limit.
