Why do fetuses have membranes between fingers and toes?
Short answer
The transient membranes between the fingers and toes in embryos are a myth.

Background
The presence of transient membranes in developing embryos supposedly points to our fishy evolutionary origins. However, I couldn't find any credible sources to support this myth. In fact, I couldn't find any source mentioning the presence of webbing between the fingers and toes at any stage during development. 

To support this lack of evidence, I have posted several photos of embryos and fetuses in various developmental stages. None show any signs of the "swimming webs". Note that the age mentioned is the developmental age, counted from the moment of conception (add 2 weeks to have gestational age):


Fig. 1. Five-week embryo. source: Palm Reading Perspectives


Fig. 2. Seven-week embryo. source: Wikipedia


Fig. 3. Eight-week embryo. source: WemMD 


Fig. 4. Fourteen-week foetus. source: WemMD 


Fig. 5. Eighteen-week foetus. source: WemMD

Fig. 2 may seem to show some webbing, but if we look at some close-ups of the hand and feet at week 7 in Fig. 6 below, we can see that there is no real webbing; fingers and toes are simply budding from the primordial hand and foot structures. 




Fig. 6. Developmental stages of the hand and feet around week 7. source: I am Pregnant

Note that an embryo is called a fetus only after the first trimester, i.e. after some 12 weeks. There is certainly not any sign of webbings in healthy fetuses. 

Also note that webbings may be the result of birth defects, such as syndactyly and pseudosyndactyly (Fig. 7):


Fig. 7. Infants with Syndactyly (left) and Pseudosyndactyly (right). sources: Wikipedia &amp; University of Columbia 

Original photographic image source (Figs 1 &amp; 5)*
- Nilsson, Ett Barn Blir Till

* Credits go to rg255