These combination contraceptives contain two substances: Estrogen and progestin (a synthetic progesterone). Both are used together to overcome the negative effects of single substance drugs as acne or sickness.
The risk for blood clots has been known for quite a while (see reference 1 and 2) and is generally caused by raised levels of clotting factors (especially VII and X), increased plasma fibrinogen, increased platelet aggregation in the blood and reduced activity of antithrombin III (which inhibits coagulation). This can happen as well in pregnancy when the factors are naturally elevated (that's the way these contraceptives simulate pregnancy to the body). The paper shows this nicely for two of the factors of the blood coagulation cascade (factor VII and X):

This risk is relatively rare, it affects only about 1-5 in 10.000 women, but is still not negligible. The risk is much higher in persons with defects in the blood coagulation cascade. Compared to this, pregnant women have a risk for blood clots of 5-20 per 10.000 women (for details see here).
Blood coagulation is really fine tuned to ensure proper function when necessary but no blood clotting when not. Basically activating and deactivating hold a balance, as depicted in figure 1 from reference 3:

Smoking now disturbs this balance by inhibiting fibrinolysis and promoting the accumulation of fibrin and pushes it more to the site of coagulation. What causes the problem here is the elevation of clotting factors in combination with the disturbance of the balance between activation and inhibition of clotting. See reference 3 for the details.
This effect seems to be limited to high concentration single estrogen and estrogen/progestin combination preparations, while there seems to be no such effect in progestin single preparations (although data is limited here). There is also a significant difference in the type of estrogen used. See references 4 and 5 for more details.
The mechanism of how this works is relatively straightforward. The estrogen binds to the estrogen receptor on the cell surface and forms a receptor-ligand complex. This leads to signalling inside the cells which finally results in changed gene expression. Additionally the complex can be internalized by the cell. Inside the cell this complex can bind to the so-called "hormone response element" and also regulate the expression of genes. This is explained in detail for Factor XII in reference 6 (and works like this for other parts as well). See the image for illustration (from reference 7):

References:
- Relation between oral contraceptive hormones and blood clotting
- Use of combined oral contraceptives and risk of venous
thromboembolism: nested case-control studies using the QResearch and
CPRD databases
- Hemostatic effects of smoking and oral contraceptive use
- Coagulation effects of oral contraception.
- Lower Risk of Cardiovascular Events in Postmenopausal Women Taking
Oral Estradiol Compared With Oral Conjugated Equine Estrogens
- Molecular basis of estrogen regulation of Hageman factor XII gene
expression.
- Estrogen synthesis and signaling pathways during aging: from
periphery to brain