How can hyperthyroidism induce osteoporosis?
T3 stimulates osteoclast formation. Osteoclasts are cells that break down bone tissue. They are essential for bone growth in children, because they allow bone remodelling. However, elevated stimulation of osteoclast formation leads to osteoporosis in adults, where bone remodelling is not necessary. 

In hyperthyrodism, T3 levels positively correlate with levels of calcium ions in blood, which confirms the break down of bone tissue. Calcium ions are then excreted in urine, because they cannot be efficiently reabsorbed. This constant loss of bone tissue cannot be reduced with high calcium diet, only by drug treatments that affect T3 levels. 

Definitely can recommend further reading, especially regarding different types of thyrodism:

DOI: 10.1002/jcp.20041