Why should a tumor look like a crab?
Such projections are more formally known as spiculations. Most commonly, we talk about spiculations with respect to the radiographic appearance of malignant breast and lung lesions. This paper* describes the correlation between the mammorgraphic appearance of spiculated breast lesions and their pathology (microscopic appearance), which is a reasonable start at addressing the "why" question you pose. The authors define spiculations as follows:


  A spiculated breast lesion is defined as a mass or an architectural distortion characterized by thin lines radiating from its margins.


Here is a picture taken from that paper which shows the mammographic appearance on the left and the microscopic appearance on the right, with the arrows pointing to the spiculations.



The linked paper gives many other examples of the microscopic and radiographic analysis of such tumors. It also explains that the two primary reasons for the spiculated shape are:


desmoplastic response
tumor infiltration (which occurs along anatomic ductal structures)


According to this report, the first explanation is more common. To understand the why behind the shape, we need to know the mechanism of this desmoplastic response. As described in the linked Wikipedia article, desmoplasia is not actually the tumor itself but dense fibrous tissue that is induced with malignant tumors invade healthy tissue. The mechanism of this is not entirely clear, but the Wikipedia article sets forth two theories:  


The reactive stroma hypothesis postulates that tumor cells cause the proliferation of fibroblasts and subsequent secretion of collagen that acts as a scaffold. This scaffold apparently forms the outline of the "crablegs".  
The tumor-induced stromal change hypothesis states that tumor cells themselves differentiate into fibroblasts and secrete the collagen that forms the spiculated scaffold.





*Franquet T, De Miguel C, Cozcolluela R, Donoso L. Radiographics. Spiculated lesions of the breast: mammographic-pathologic correlation. 1993 Jul;13(4):841-52.