How does fluorouracil inhibit thymidylate synthase?
The inhibition comes about because:


Uracil and $5$-fluorouracil are sufficiently similar that both moieties have a strong binding affinity for the active site of thymidylate synthase.
The presence of the fluorine prevents $5$-methylation of the uracil core by a folate cofactor, which otherwise would turn it into a thymine moiety.
This inability of the enzyme to convert the $5$-fluorouracil moiety into a thymine moiety (which has a much lower affinity for the active site) means that the enzyme has no way to reject the $5$-fluorouracil or the methylating folate species, once bound.


The particulars of this interaction are nicely spelled out in Mishanina et al.$^\dagger$, conveniently available in preprint:


  [5-fluorodeoxyuridylate] reacts [as normal] with the active site cysteine and undergoes condensation with [N,N-methylene-tetrahydrofolate] but prevents proton abstraction and elimination of the tetrahydrofolate [product], as a consequence stalling the reaction at the covalent ternary complex [of enzyme, inhibitor, and folate species].


Scheme 4A from the preprint linked above illustrates well the specifics of the chemistry.  (I refrain from reproducing it here due to uncertainty regarding its copyright/license status.)

As well, Figure 2 from the Mechanism Description section of the Wikipedia page on thymidylate synthase shows very nicely the enzyme interactions with the normal deoxyuridylate substrate (click image for larger version):

$\hspace{20pt}$ 
Image by CJerc (Own work) [CC BY-SA 3.0 (http://creativecommons.org/licenses/by-sa/3.0 )], via Wikimedia Commons

$^\dagger$ Mishanina et al. Bioorganic Chemistry 43: 37 (2012). doi:10.1016/j.bioorg.2011.11.005