Hi Caty,
Your question could be rephrased to say: why does molecular biology work? It is no small thing, which is why the discovery of one of the most famous reporter genes, GFP, was awarded the Nobel prize last year.
The main concept to consider, as Shubjangi pointed out, is the way in which you put together your construct. Essentially you are tricking the cell into thinking that it is expressing something that is normal. For example a lot of constructs use a constitutive promoter that has all the necessary characteristics to be recognized by the cell's machinery so it is used. It is almost like driving a car in the middle of San Francisco that happens to have the steering wheel on the right side. It is "completely" different from all the other cars, but as long as you know the rules of the road you can drive around without any problems (of course, assuming that there are no laws against doing this :) ).
As for your question about a cell allowing the expression of genes within itself: it depends on which cell type we are talking about. If you are a prokaryotic cell, then you have a mechanism to recognize these foreign transcripts and degrade them using restriction endonucleases. If you've expressed proteins in bacteria you are likely aware that there are commercial competent cells out there that have been engineered for protein expression. This was achieved by removing the machinery within these bacteria that recognizes foreign transcripts (and proteins) so they are not degraded. In eukaryotic cells, like human or mouse cells, since they come from a multicellular organism with its own immune system, these individual cells do not have the machinery to recognize foreign genes. As a result, for the most part you do not need to engineer these cells since you can express whatever you want in them and they will not be aware something foreign is present. You could say that eukaryotic cells, unlike wild type prokaryotic cells, are blind to being modified genetically. In many ways, this is the price eukaryotic cells pay for being part of a complex multicellular organism (one of the many reasons why viruses are able to infect us early on before our immune system kicks in).
Cheers