If my paper is accepted in a Springer journal, can I submit my version of the manuscript to arXiv immediately after acceptance?
The short answer is: yes, staggered posting rights like this are very common - many journals distinguish between your own website, your institutional repository, and a broader repository (eg arXiv, pubmed), with different rules about what version of the article can be posted and when; some have also begun to provide a special category for sites like ResearchGate. 

The underlying answer -

Why on earth do they do this?

On the one hand, this doesn't make sense - the PDF is the PDF and once it's picked up by a search engine, it's going to be readable wherever it's hosted. But consider it in terms of discoverability and scale:


one hundred papers from the journal on one hundred obscure personal sites;
one hundred papers from the journal on twenty well-organised institutional sites; or
one hundred papers from the journal on one disciplinary repository


From a publisher's perspective, the first option is not something to worry about, while the second &amp; particularly third options look quite scary. Remember, what they really don't want is everyone saying "great, we can get this all from arXiv, cancel next year's subscription please". So an embargo period gets attached to the repository copies - many journals (most prominently PNAS) manage fine on a subscription basis while still making older papers freely accessible after a year or two, and so it's well-understood that allowing delayed access in one form or another will not ruin the subscription income.

Now, mass cancellation because of self-deposited papers being available instantly is a bit of a bogeyman. No-one's really shown it would happen on a large scale (and indeed arXiv suggests otherwise); library budgets are not (yet) squeezed enough that we've had to start thinking seriously about it; and in any case "big deal" subscription models often make it impractical to cancel specific titles. But it's looming as the threat and most publishers simply don't want to risk it... so they produce very conservative guidance on what you're allowed to do, and work from there.

Over the past few years, many of these publisher limits have loosened slightly as they discover the sky didn't fall, which I suppose is something. At the time of writing, at a very loose generalisation, policies for non-medical scientific journals are mostly converging along the lines of "accepted MS immediately on your own site, accepted MS in a repository after a year, publisher/proof PDF never", but there are a thousand variations.