Do buddhists fall in love? I'm new to the concept of Buddhism, and am failing to understand perhaps something very basic. I understand that one is not expected to hold anything close or dear as it is impermanent and could change at any moment so how does one love another person? How does one become close to another person. As I understand it, a relationship would just consist of two people who aren't that close to one another.

"Therefore hold nothing dear, for separation from the dear is painful" - this seems to suggest that you should build a wall around yourself and separate yourself from potentially wonderful feelings just because they could cause pain, and pain is bad. 

I think pain is a part of life; denying that pain, or ignoring it is not part of a healthy life. I think of this concept of Buddhism similar to "don't smell the flowers because you may prick yourself on a thorn" thus denying yourself of the delights found in the smell and also the growth / knowledge from the pain of the thorn prick. Both of which you can learn and grow from.

Am I missing something here?
It depends which Buddhism we are talking about:

Both Theravada and Mahayana, including Zen, would consider romantic love a disease of the mind, a kind of pathological obsession. Theravada would offer the more universal emotion of metta instead -- the loving-kindness, although in Theravada it is usually applied as an antidote against hatred, for the benefit of one who experiences it. Most schools of Mahayana would sincerely rejoice in the selfless aspect of love. In Mahayana one is supposed to sacrifice one's interests for the benefit of others. Dropping one's ego is Mahayana's path to Enlightenment.

In Vajrayana schools though, emotions, including romantic love, are considered a form of energy that can be put to use, or at least accepted as part of the given. Vajrayana perspective would more likely align with your "pain is part of life" and "might as well enjoy the flower" conjectures. With one caveat though: Vajrayana would still appreciate the inherent fakeness of love, the mechanical nature of which comes from a match of partners' stereotypes and preconceptions. So even if a Vajrayana practitioner could play with the fire both in context of its ego-melting properties as well as for pleasure, they would not take it one-sidedly as an untrained run-of-the-mill person would do.

EDIT: what I meant by "Vajrayana practitioner ... would not take it one-sidedly" -- I meant that Vajrayana view includes both sacred and illusory aspects of love. In Vajrayana we are trained to see things from all the sides at the same time. Love is both sacred and a giant trick, as far as Vajrayana is concerned.

The predominant Buddhist sentiment here is that being disappointed/disenchanted (="sober") is a helluva lot healthier state than the state of intoxication by an object of mind. While Vajrayana is 100% aligned with this most fundamental of Buddhist principles, we do allow ourselves to get drunk, both metaphorically with love, and occasionally even literally - while staying fully accountable for the consequences - a trait of the universal adult.