First, I would like to say that there is no reason to believe that this is thyroid cancer. Hypothyroidism, anti-thyroid antibodies, and thyroid cancer can all be completely independent of eachother, although having antibodies against your thyroid can result in hypothyroidism (or in some cases hyperthyroidism).
The most common cause of hypothyroidism in countries who add iodine to salt is an autoimmune disease called Hashimoto's. Hashimoto's is often diagnosed with anti-thyroid peroxidase and anti-thyroglobulin antibodies. Since you have high anti-thyroid peroxidase antibodies, you most likely have Hashimoto's (I do too). (I just looked up my original test from 2018 - the range for my test was less than 250 is normal, over 250 is high. my antithyroid peroxidase was 1000 - high).
I am very surprised your doctor put you on iodine, unless there is some reason to suspect thyroid problems caused by iodine deficiency (and again, this might depend on where you live/whether your salt is iodized or not). Unfortunately, if you have autoimmune hypothyroidism, taking iodine probably will not help, and possibly could exacerbate your symptoms. Autoimmune hypothyroidism occurs when your body develops antibodies that attack the thyroid. If this does not affect your thyroid hormone/cause symptoms, your thyroid hormone levels can be monitored but you might not need treatment right away. Hashimoto's is often a progressive disease though, and as the disease progresses, more and more thyroid tissue is destroyed by the immune system. Since you already have a somewhat elevated TSH and have symptoms, there is a good chance you would benefit from supplementary thyroid hormone (for example, levothyroxine or natural dessicated thyroid).
If you would like to report your T3 and T4 test results with their reference ranges, we can help you figure out if those are also low. Most people feel better/have fewer hypothyroidism symptoms if free T4 and free T3 are in the top half of their ranges, so even if it is "normal", it might be too low for you.
I'm not at all familiar with hypothyroidism caused by iodine deficiency since that is not what I had, but if it were me, I would ask the doctor about the possibility that you have Hashimoto's/an autoimmune thyroid condition, and what type of treatment is best for that, because the treatment is very different from iodine deficiency hypothyroidism.
What would cause the antibodies to be high? Good question. I know women are much more likely to have Hashimoto's than men, and the risk of developing Hashimoto's increases with age (I was diagnosed at 37). There also seems to be higher prevalence of autoimmune thyroid disease in countries that do iodize salt. Why people develop autoimmune conditions in the first place is an area of active research right now, and I don't have any straight forward answers to that.
Finally, I have zero reason to believe you have thyroid cancer, but since you possibly have Hashimoto's and hypothyroidism, it would be worth getting a thyroid ultrasound to make sure you do not have suspicious nodules. Hypothyroidism can cause an increase in TSH, which is a hormone the pituitary gland secretes that tells your thyroid to "grow, grow, grow". Chronic, untreated Hashimoto's can result in a multinodular goiter (which is what I had - by the time I figured out what my problem was, my thyroid was very unhealthy). Thyroid nodules are often benign, but sometimes can be cancerous, so it might be a good idea to get a thyroid ultrasound. I was told by my ENT that 30% of women in their thirties have nodules, 40% of women in their 40s, 50% of women in their 50s, etc. have nodules, and the vast majority of those are not thyroid cancer, so even if they find nodules on your ultrasound, it does not mean thyroid cancer. So... I would not suspect thyroid cancer, you can have hypothyroidism and no nodules, or nodules and no thyroid hormone problems at all, but it is worth making sure both are being looked at and monitored since you now know you have hypothyroidism.
Hope this helps,