I must say, that I like this idea and it seems that it will bring a new form of protection against CSRF and XSS or at least it will reduce those attacks.
So, how effective will this protection be?

SameSite-cookies is a mechanism for defining how cookies should be
sent over domains. This is a security mechanism developed by Google
and is at this moment present in Chrome-dev (51.0.2704.4). The purpose
of SameSite-cookies is [try] to prevent CSRF and XSSI-attacks. You can
read the draft here. (Source)


First, a definition from Chrome:

Same-site cookies (née "First-Party-Only" (née "First-Party")) allow servers to mitigate the risk of CSRF and information leakage attacks by asserting that a particular cookie should only be sent with requests initiated from the same registrable domain.

So what does this protect against?
CSRF?
Same-site cookies can effectively prevent CSRF attacks. Why?
If you set the session cookie as same site, it will only be sent if a request emanates from your site. So a standard CSRF attack where the attacker lures the victim to the site http://malicious.com that posts a request to http://bank.com/transfer.php?amount=10000&receiver=evil_hacker will not work. Since malicious.com is not the same site as bank.com, the browser will not send the session cookie, and transfer.php will execute as if the victim was not logged in.
This is very similar to how setting a X-Csrf-Token HTTP header protects you from CSRF - again there is something that is only sent if the request emanates from your domain. The same-site property turns your session cookie into a CSRF-token.
So problem solved? Well, sort of. There are some caveats:

This will only work for browsers that implement this feature. So far, very few do. Unless you want to throw everybody who uses a slightly older browser under the bus, you will still need to implement an old fashioned CSRF-token.
If you need more fine-grained control, this will not be enough. If you run a system that displays untrusted user content, like a forum, you don't want requests originating from user posts to be treated as valid. If someone posts a link to http://myforum.com/?action=delete_everything you don't want that to delete anything just because a user clicks it. With a traditional CSRF-token, you can control exactly when it is used. With a same-site cookie, you can not.
The same old caveats as for old fashioned CSRF protections still apply. If you have an XSS vulnerability, no CSRF protection in this world will save you.

With that said, the same-site cookie is still a good thing that should be used together with traditional defenses as defense in depth.
XSS and XSSI?
The same-site cookie does nothing to protect you from ordinary XSS attacks. If a hacker manages to fool your site to echo out script from the URL on your site, it will be executed as coming from your origin (after all, it is), and thus session cookies will still be sent with all requests the injected script makes to your domain.
If you read the quote in your post carefully, you will see it says XSSI with an I at the end, and not XSS. The I stands for inclusion, and XSSI is related to, but different from, XSS. Here is a good explanation of XSSI and how it differs from XSS:

XSSI is a form of XSS that takes advantage of the fact that browsers don't prevent webpages from including resources like images and scripts, which are hosted on other domains and servers. [...] For example, if Bank ABC's site has a script that reads a user's private account information, a hacker could include that script in their own malicious site (www.fraudulentbank.com) to pull information from Bank ABC's servers whenever a client of Bank ABC visits the hacker's site.

Same-site cookies protects you from XSSI, since a session cookie would not be sent with the request for the script and it would thus not return any sensitive information. However, for ordinary XSS it makes no difference.
