Let us consider the example of "no less than the country's president attended".
In many countries with the title president, they are of considerable political power. In most of the rest, it is still a position of considerable esteem. As such, there would be few, or no, potential attendees of greater rank.
So, the set of people who are "not less" than the president will be very small; either just the president or else the president and a handful of other people.
To have someone from that small set attend is therefore remarkable.
So, it literally means "someone who ranks as least as highly as the president attended", and the person referred to in this roundabout fashion is indeed the president. You would not expect someone to say "No less than the president attended. It was the Taoiseach who has more real power".
This is a from of litotes where we use an understatement to produce emphasis, often a double-negative or negating a comparison to produce the opposite comparison as in this case ("no less than" meaning "at least as great as").
Litotes is found in many languages, and very common in English. We have "not bad" for "good", "not bad looking" for "attractive", "no spring chicken" for "old", and so on. English has quite the history with it, as Old English poetry used it a lot.
It can perhaps be best read as "Wow, the president attended, can you imagine, the president. That's the most impressive person there could possibly be attending!", but from a more reserved speaker.