My experience of coaching swimmers who are complete new to TI is that the first major improvement you often see is the swimmer achieving a sense of balance in the water. By allowing the water to hold their body this keeps them in the best and most comfortable position for swimming. This also helps to answer the question: "but why are my legs always sinking?". TI drills like Superman Glide and Skate help to introduce this feeling of being supported by the water much faster than trying to correct an imperfect full stroke. TI swimmers also find with practice that maintaining a more relaxed head and neck held by the water also reduces the aches and strains that they had when previously swimming freestyle.
What is more difficult to teach, and is a disappointment for some swimmers starting out, is that it is not through swimming 100s of lengths/laps, or very quickly covering great distances (5k, 10k etc.) that your swimming technique is going to radically improve. There might be some improvement - no doubt merely by time spending time in the water - but this approach can also make worse bad habits that are already there and that need to be worked out in finer detail. And whilst there is definitely a place for aerobic fitness in TI swimming it is not at the start of the training as high up the list as improvements to technique and mental attitude.
Why is it better to practice in short sequences, like 6 to 8 strokes? - The importance of maintaining focus while developing a new skill is essential to achieving mastery. Swimmers, especially when they are starting out, will find quickly that distractions (things around them - the need to take a breath - other parts of their stroke) quickly interrupt their focus when swimming. I always recommend to my clients to spend at least some of their swimming time in a shallow practice pool, or swimming only in the shallow part of a pool, so that they are also not distracted by the waters depth and can stop and stand whenever they need to. It's easy to think when you have been swimming a long time that "this is kids stuff" and surely deep vs shallow water makes no difference. But I can guarantee that whichever stage you are with your swimming being able to stop and stand in shallow water will greatly help increase your focus on other things.
Another timesaver to improve your technique is to go swimming with a plan. Nearly every time I go swimming (and I recommend my clients to do the same) I take a waterproof notebook and pencil to record exactly what I am practising. This might sequences of drills, focal points or tempo-trainer practice. I've now developed a kind of shorthand so this: 4 x 6S FP L + R ragdoll arm' means 'practising four times 6 strokes with the focal point of keeping a relaxed ("ragdoll") recovery arm on left and right'. (I'd recommend coming up with a shorthand method that makes sense to you).
If you are worried that things like practising drills in the shallow end or writing up notes between lengths / drill sequences will make you stand out from other swimmers at the pool - just remind yourself that the quickest and easiest way to become a great swimmer is not churning up and down the pool without structure or focus. You are working on something important by practising these things. TI helps you to learn to master swimming in a systematic and focused way, you might even find this sense of focus and mindful practice helps you to learn things away from the pool too!