Why is quantum entanglement considered to be an active link between particles?

Entanglement is being presented as an "active link" only because most people - including authors of popular (and sometimes even unpopular, using the very words of Sidney Coleman) books and articles - don't understand quantum mechanics. And they don't understand quantum mechanics because they don't want to believe that it is fundamentally correct: they always want to imagine that there is some classical physics beneath all the observations. But there's none.

You are absolutely correct that there is nothing active about the connection between the entangled particles. Entanglement is just a correlation - one that can potentially affect all combinations of quantities (that are expressed as operators, so the room for the size and types of correlations is greater than in classical physics). In all cases in the real world, however, the correlation between the particles originated from their common origin - some proximity that existed in the past.

People often say that there is something "active" because they imagine that there exists a real process known as the "collapse of the wave function". The measurement of one particle in the pair "causes" the wave function to collapse, which "actively" influences the other particle, too. The first observer who measures the first particle manages to "collapse" the other particle, too.

This picture is, of course, flawed. The wave function is not a real wave. It is just a collection of numbers whose only ability is to predict the probability of a phenomenon that may happen at some point in the future. The wave function remembers all the correlations - because for every combination of measurements of the entangled particles, quantum mechanics predicts some probability. But all these probabilities exist a moment before the measurement, too. When things are measured, one of the outcomes is just realized. To simplify our reasoning, we may forget about the possibilities that will no longer happen because we already know what happened with the first particle. But this step, in which the original overall probabilities for the second particle were replaced by the conditional probabilities that take the known outcome involving the first particle into account, is just a change of our knowledge - not a remote influence of one particle on the other. No information may ever be answered faster than light using entangled particles. Quantum field theory makes it easy to prove that the information cannot spread over spacelike separations - faster than light. An important fact in this reasoning is that the results of the correlated measurements are still random - we can't force the other particle to be measured "up" or "down" (and transmit information in this way) because we don't have this control even over our own particle (not even in principle: there are no hidden variables, the outcome is genuinely random according to the QM-predicted probabilities).

I recommend late Sidney Coleman's excellent lecture Quantum Mechanics In Your Face who discussed this and other conceptual issues of quantum mechanics and the question why people keep on saying silly things about it:

http://motls.blogspot.com/2010/11/sidney-coleman-quantum-mechanics-in.html

I wish to complete @Luboš Motl's answer, to which I agree. My point is on why people continue to make this mistake of an active link. This mistake is connected with one of the most interesting properties of quantum mechanics, Bell's theorem. One can argue that any physical theory is an hidden variable theory, the hidden variable being the description of the state of an object as written by the theoretician describing it. For quantum theory, the wavefunction of the object is the hidden variable.

Bell's theorem state that the prediction of quantum theory cannot be described by any local hidden variable theory. More precisely, for any entangled state, you can find a set of measurement with statistics contradicting any local hidden variable theory. The three possible explanations are:

1. Nature is not local : your physical description is a real physical object, and there is an active non-local link between the two entangled particle.
2. Nature is not realist : your physical state is only an approximation and has no real meaning.
3. Nature is not quantum.

(1) is much easier to explain and appears often in popular science, mainly because (2) is much more difficult to explain and accept. But I think most researcher working with entanglement prefer explanation (2). Einstein intuition was 3 (before Bell's theorem), because he could not accept (1) and (2).

Interestingly, Einstein 1936 original paper on the EPR paradox was on a case where you can easily find a local hidden variable theory. The state described it what is now called a two-mode squeezed state. Its Wigner function is positive and can therefore be interpreted as a classical probability distribution on the quadrature (position and momentum) measurements, the only one discussed in the EPR paper. Such classical analysis of entanglement can be theoretically very useful and help the intuition in some case without needing any spooky action at distance. However, as shown by Bell, such local hidden variable theory cannot be generic enough to encompass all quantum mechanics.

Just a nice analogue Prof. Jürgen Audretsch told me once:

Imagine at home you put one glove in your coat without looking (and noticing it's only one of the two). After exiting the train you notice it's cold and you pull out that single glove. At this very instant you know it's either the left or the right glove, and you therefore know which one is left at home. However, no information was transmitted by your "measurement". Of course in quantum mechanics this is more complicated because of the not entirely measurable wave function, but this is the basic idea.