TL;DR
1. You will not pay taxes in the USA. 2. You will pay income tax in India as per 44AD/44ADA or on actual profits. 3. You will be treated as an exporter for the purpose of GST.
Why no US taxes?
A single-member US LLC is a 'disregarded entity' under US tax law. The IRS does not tax the LLC separately — income passes through to the owner. Since the owner (you) is an Indian tax resident and not a US person, the LLC has no US tax filing obligation for the pass-through income. However, you may need to file Form 5472 for reporting purposes.
Indian tax treatment
Under the India-US DTAA, business profits are taxable only in the country of residence unless there is a Permanent Establishment in the other country. Since your PE is in India (you work from India), all profits are taxable in India. You can file under Section 44ADA (50% presumptive profit) if your receipts are under ₹75 lakhs.
GST treatment
For GST purposes, receiving payments from a US LLC that you own does not change the analysis. You are providing services and receiving payment in foreign currency. You are an exporter of services. File your LUT and charge 0% GST. The invoices should be raised by you (the Indian proprietor) to your clients, not from the US LLC to you.