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I choose an airline that allows ESA animals. I, or a travel organization, contact the airline of my choice.
On the website normally there is the item “Traveling with an animal”. I read the relevant provisions carefully. I register my pet.
In addition, I contact the airline by e-mail and have me confirmed in writing that my ESA animal can fly with me in the cabin. The confirmation must include details of the animal, my person and all flight dates and times.
In case of doubt, I can blame the airline for accepting liability, e.g. in case I incur costs due to delays. All necessary documents must be drafted in English or the national language of the destination country.
Experience has shown that ESA animals are relatively easy to accommodate when travelling to/from or through the United States. This especially when I book the flight with an US airline. Background is the legal basis for the function of ESA animals enacted in the USA.
Please note, however, that each airline is entitled to reject or re-book animals for safety reasons if necessary.