In a hay infusion, how does bacteria appear in the first place?

Spore forming bacteria like Clostridium or Bacillus. Spores are resistant to boiling temperatures that kill vegetative cells. After the hay infusion was boiled, you cooled it down and then the spores germinated and began growing as vegetative cells.

You need to autoclave the hay to sterilize it.

Pasteur preformed many experiments to demonstrate that living organisms do not appear unless they are already present. An autoclave is a sealed vessel in which water is heated above its boiling point, normally to around 240 degrees F. The pressure increases to around 15 PSI and after 20 minutes, there will be no micro organisms alive. Many micro organisms can survive ordinary boiling water at atmospheric pressure. Also, if a container is not covered, dust will settle on the surface and introduce quite a number of bacteria and fungus. This is why a stewpot has a lid. If cooked and then covered every evening, refrigeration is not needed. Home canning employs a pressure cooker. Food is packed into glass jars and the caps are placed on top but not tightened. This prevents the jars from cracking inside the cooker if the pressure rises too quickly. Wax can be added, which will melt and form an additional barrier. After half an hour, the heat is turned off. When the pressure cooker returns to room temperature, it is opened and the lids tightened on the jars. Done properly, home canning is as efficient as metal canning.