The saying curiosity killed the cat is in fact based on the opposite reason for cats exhibiting such high levels of curiosity – curiosity in fact keeps them alive. Cats are highly intelligent and have great memories coupled with an interpretive mind. To survive in the world the need to make sense and analyse the surroundings environment. They need to be able to determine what is potentially dangerous and what is potentially food.
Cats are highly territorial – unless pushed out by a stronger cat they won't willingly leave theirs once it's been established. In order to get the most out of their territory they spent a lot of time exploring it and mapping it. They map the places where they can rest safely, where they can find food most easily in the areas that are dangerous and to be avoided.
Major changes in their environment can affect all of these things and thus they need to be investigated to determine the impact. Expect any new additions to their environment, better children's toy, a box, the shopping that you've just brought home to require immediate inspection. A major change in your cats environment such as completely remodelling a room with furniture in different places and new furniture added will require hours or even days of inspection and assessment. Yes your cat might indeed like the new couch but they're rubbing on it is more likely scent marking as opposed to a display of affection. Whilst few cats enjoy driving in cars as this means involuntarily leaving their territory they do like to inspect them softer car door has been left open and your cat has gone missing and certainly worth checking under the back seat.
Whilst your cats' natural curiosity is a survival mechanism it can certainly land them in danger. Is that electrical cord something to be played with, something to be hunted or something to be cautious of. Unfortunately, there are some circumstances where the natural investigative process of the cat is very likely to lead them into danger such as investigating slippery areas were a fall could lead to injury, tasting poisonous items et cetera.
Domestic cats get less curious as they get older partly because they are more certain of what is going on in their environment and less things need investigation but also because they become accustomed to being provided form so their natural curiosity dims. A cat that knows it will be fed morning and night doesn't need to be too curious about the best hunting grounds in its territory.
expect your cat to regularly patrol their territory and take note of changes or evidence of incursions into their territory by other cats. Expect them to pick up on the scent of other cats, to notice evidence of potential prey such as birds, mice or insects that have moved in. All of these changes need to be assessed to determine whether they are either threat or food.
Cats that spanned much of their time inside may still consider the area around the house that they can see to be their territory and so they will spend time on windowsills surveying all that is around them. It makes no difference that your Can have no impact on the things that are happening outside… Birds flying past, other animals other people et cetera. These are still all things that need to be inspected and noted.
Expect your cat to recognise items that have historically come and gone from its territory. Shopping bags often mean a refilling of their favourite food. A Travel basket might mean an unwonted leaving of their territory to visit the local vet so be careful when bringing out such an item if you don't want to spend the rest of your day looking for a hiding cat.
when your cat is on patrol, checking its territory to change new additions confirming it's remembered best places for hunting sleeping et cetera remain in place… they generally do not wish to be observed. Whilst the dog will run around barking and inspecting their environment noisily your cat will be more like an undercover agent. Quiet, remaining still and lying low for long periods, avoiding being observed themselves.
This of course makes perfect sense since the major reason for this inspection is potentially fine prey or locate danger in both cases look at does not want their target to notice them.
When your cat is on patrol it's very likely they will not be receptive to a pat or may ignore all the sound of your voice calling them. Whilst this is often wrongly misinterpreted as aloofness it really is just a cat on patrol doing what cats on patrol do best. Staying focused and alert, not giving away their position and paying close attention to the important unknown stimuli whilst ignoring the known nondangerous sounds from their human.