Is that correct that I will not do any violations doing so [with the YouTube Standard License]?
The YouTube Standard License basically authorizes YouTube to distribute the video via its platform. Remaining rights not covered by the license are retained by the original author(s) of the video.
As @StephenOstermiller points out, YouTube's embedding widget is specifically designed for sharing via YouTube, but on a separate web page.
So using this widget to display the video on your page would be perfectly acceptable.
However, if you simply downloaded the video and embedded it in your page via another player (not YouTube's widget), you would almost certainly be running afoul of YouTube's licensing and likely general copyright law depending on where you live (since only YouTube and the original author have distribution rights).
Note: Even with the widget, playback via another web page may be disabled. If this is the case, the widget will display a message saying so and direct viewers to the original YouTube video.
Is it OK to skip asking permission from video owner?
If you use YouTube's widget, yes. Even so, linking back to the author is still a nice gesture in my opinion.
The other case for not asking permission would be if the video is under a Creative Commons License (this is the other license authors can use for their work on YouTube).
Am I correct that this will really improve my traffic quantity and quality?
Maybe. It depends on how Google, et al. decide to treat the video.
As a for instance, certain queries in Google will bring up StackExchange listings (which do not have videos) before other results (including YouTube video tutorials).
My personal thought would be to embed the video if it enhanced the user's experience. If it gets a higher ranking, great. It is extremely unlikely to hurt your ranking in any meaningful way, regardless.