First of all, it's important to distinguish between the types of fatty acids that may form the cell's membrane:
Note that both cis- and trans-fatty acids are unsaturated, the difference being the conformation around the double bond. A trans-fatty acid is more-or-less a straight line, whereas the cis-fatty acid has a noticeable "kink" at the point of the double bond.
Now, the suggested molecular mechanism behind the health risks associated with trans-fatty acids has to do with a property of the cell's membrane called membrane fluidity. This property affects the cell's ability to send and receive signals, and it greatly affects the function of many proteins whose structure and role are dependent on the membrane's state. The membrane has to be fluid enough in order to serve its purpose properly.
(If you don't have time to read a lengthy answer, you can skip to the last paragraph now. Otherwise, keep going...)
One of the main functions of the cell's membrane is to separate the inside of the cell from the outside world, in order to control the cell's internal environment, help it avoid damages from the surrounding environment and deal with harmful substances or unfavorable living conditions. However, the cell needs to communicate with the outside world through different signaling mechanisms which involve sending and receiving chemical signals, as well as adapt itself to different living conditions.
In order to do that, the cell has to have a membrane with a dynamic structure, i.e. one that will be able to adjust itself to the surrounding environment. Evolution decreed that this may be best achieved by a lipid bilayer.
This image shows a portion of a cell membrane, which is made of many fatty acids in a formation that is considered to be the most stable energetically, due to the hydrophobic/hydrophilic interactions of the fatty acids among themselves and with the surrounding water. This image also shows how cholesterol is embedded here and there within the bilayer.
The measure of a membrane's fluidity results from the type of fatty acids that form it and the membrane's cholesterol content.
To put it simply, the more straight fatty acids in the membrane, the more rigid (i.e. less fluid) it will be. The more "interruptions" to the hydrophobic interactions between the fatty acids' tails (by "kinks" in the tails or cholesterol), the more fluid the membrane will be.
The above picture shows two distinct states of the membrane. On the right-hand side the membrane contains mostly straight (i.e. saturated and/or trans) fatty acids, which results in a more rigid and viscous membrane. This state makes it more difficult for the cell to communicate with its surroundings (chemical signals cannot enter the cell easily, proteins in the membrane are less mobile, etc.).
On the left-hand side, the membrane contains more unsaturated fatty acids of the cis conformation. The "kink" that characterizes cis-fatty acids is the main reason this membrane is more fluid - it makes it harder for the fatty acids' tails to stick together due to hydrophobic attraction.
Cholesterol serves the same purpose as those "kinks". by disrupting the attraction between fatty acids' hydrophobic tails. This is a very important role of cholesterol in virtually all of the cells in the body, which makes cholesterol a very important molecule in the body.
To summarize, trans-fatty acids are considered unhealthy because when consumed, they will mostly remain intact (i.e. will not be metabolized; as this form of fatty acids is not found in nature, the body's enzymes have a limited ability to catabolize them), and then they can penetrate cellular membranes, form rigid hydrophobic bonds with the other fatty acids and decrease the membrane's overall fluidity.
See Diet and HD - article from Stanford university.
Other references may be found in most introductory biochemistry texts or online (be sure to read only scientific texts from reliable websites such as the one given above).
