Though the abstract concept of infinity is a very intriguing idea, with little bit of effort, a bit of examples and logic, one can grasp the notion.
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The usual place where infinity is discussed is space. Space is infinite. when such a statement is made, it is not made quietly. it is a progressive conclusion, that begins with the here and now, and goes beyond the boundaries of the town, the city, the state, country, continent, the planet, solar system, galaxy and so forth. each one is an order of magnitude larger. This introduction of infinity is indeed mind boggling. In fact mind numbing. Obviously this results in the incomprehensible nature of infinity. The simple statement infinite is endless, is accepted for its literal meaning by most, and is left at that.
But an inquisitive mind, is never at rest. For instance, the progression in the expansion of space is very clearly accessible to the human perception. one can see the house, the street, the town, can travel the country, can travel around the planet... and this teasingly possible nature, also creates an ultimate desire to perceive infinity. This perpetual desire, also creates an hopelessness when one keeps on learning the expanding size of the universe, and the impossibility of perceiving the infinity.
The other approach to infinity is through abstract notion of numbers. There are infinite numbers. There is a progression from small numbers to larger and larger numbers , or smaller and smaller numbers, as the case might be. Here again, the act of symbolizing numbers, makes it teasingly a possibility to eventually 'count' infinity.
This problem of grasping infinity by the perception, either through the physical or the mental faculties, is somewhat circumvented by merely simplifying infinity, into a logical notion. "There is always a number after every number". Or in terms of space, space is never ending, so "once you come to an apparent 'end', you always see a further point".
The real solution to the impossibility of the perception of infinite lies in the fact that the physical reality is not infinite. Infinity does not exist.
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Its a mere fabrication of the mind. What you see, what you perceive, what you 'count' is all there is. So in that sense you are perceiving infinity. What you see, what you think, what you know, is all there is, and always so will it be! The notion of time is an illusion, because one can only perceive the present. Past and future are perceived as notions only in the present.
So, the endlessness of space is possible only when we travel in space AND time. There must have been a time when we reached and saw only so much, say point A, and then at another time we pursued and found another further point B. But there were no two points in time to say, "we saw point B after A, and this seems to go on forever and hence space is endless".
Space, time and for that matter anything in this universe is finite, and what we perceive is all there is.