How can I calculate the load capacity of a platform?
Here would be my basic approach (Mechanical Engineer here, Statics TA for 4 semesters):


For starters, you could figure out the weight of the mattress and box spring plus the weight of two people lying on it (W). Add in a safety factor (at a minimum 2, ideally a bit more) - remember, an uneven or dynamic loading will apply significantly higher stresses to your frame.
If you assume the weight will be evenly distributed it makes things a whole lot easier.
Figure out which supports your box spring will be resting on. Does it come in contact with the outer skirt on the sides? If not, divide up the weight of the box spring/mattress/people by the number of cross supports (n) the entire thing rests on, less 1 (W/(n-1)). For the end cross supports, use half of this loading.
Use the distributed load (or half of it) that you just calculated over the span of each cross support in conjunction with a fixed-fixed cantilever beam equation explained in Figure 23 here.
Calculate maximum beam stress at the center with σ=M*c/I (where M=moment,c=distance from neutral axis, and I=area moment of inertia) and compare to your material (pine? oak? maple?) maximum compressive and tensile strengths.
Apply the reaction forces for the cross piece ends to the sides of the supports running the length of the bed as point loadings. (If you want to get fancy, you can include the moment as well and add a torsional stress to the beam, but I don't think I can explain how to do that very well over the internet.) Use superposition (multiple P forces) and Figure 25 for this one. Repeat calculating maximum stress.


If you end up with not enough of a safety factor for your chosen design, increase the width (vertically) of the side beams. For the center, you could just add another support and recalculate. I don't think fasteners will be the limiting factor here, and calculating/estimating the stress concentration factors in an anisotropic material such as wood is a bear.

Edit: So that's how an engineer would approach this. However, @rachet freak has a very good point that you could save a lot of time by using a proven design. In that case, take a look at Ana White's website. It's got dozens of plans that people have already built with basically just a circular saw and a drill.