"Streng" or rigorous stands for what strictly divides Husserl from the natural sciences, including social sciences that imitate mathematical physics in methods. On the one hand, Husserl militates against Marburg Neo-Kantianism. which phenomenology will eclipse: on the other, he opposes the Berliner Dilthey´s historicism for its scientism. Phenomenology, or "rigorous" science, substitutes "essential seeing" for "seeing with the senses."  Hence it is purer (in a Platonic sense of apriority).  

Among the rigorous parts of philosophy - perhaps we can call it theoretical philosophy - we count logic and the theory of science. It is definitely deeper and in a sense more rigorous than much math, as it asks, for example, questions about the foundations of mathematics and logic. Mathematicians tend to go with the axioms and other foundations as they are - philosophers may ask "why?" and may question these foundations. I have both read some and been to seminars in the field, and believe me - it is seriously deep. :-) 

The little text (Philosophie als strenge Wussenshaft) published in 1910/11 stands right in the middle of the process in which Husserl is moving toward his Ideas (I, II, and III) over the following years. Properly speaking it stand in very center of the process in which Hussel moves from the static on to the genetic phenomenology.

It is a beautiful programmatic text in which Husserl anticipates the need to step aside of the "current" philosophy and science, aiming at his own phenomenologuical philosophy.

Implicitly the call for the change of attitude from the Natureinstellung on to the phenomenological one can be seen in PSW.

The rigor means both a suspicioon of what goes without saying, of everything that is taken for granted, and the need for a personal grounded reflection.

For Husserl the rogor of philosophy certainly has a sense of rigor like mathematics, and yet, this is not to be taken as if Hussserl was taking math as a model for philosophy. Far from Plato, hence.

In his IDeas I he will clarify such a elationship with one marvellous concept: ideation. It is, namely the call for the freedom of imagination in order to grasp trhe as many facets of an object as possible.

"Streng" or rigorous stands for what strictly divides Husserl from the natural sciences, including social sciences that imitate mathematical physics in methods. On the one hand, Husserl militates against Marburg Neo-Kantianism. which phenomenology will eclipse: on the other, he opposes the Berliner Dilthey´s historicism for its scientism. Phenomenology, or "rigorous" science, substitutes "essential seeing" for "seeing with the senses."  Hence it is purer (in a Platonic sense of apriority).  

To be honest: The explanation is given in the first sentence.

Science, which suffices highest theoretical needs and allows with regard to ethical-religious a life that is regulated by pure rational norms.

So the claims are not very different from where we see philosophy today: A bridge between natural sciences and religion... or lets better say metaphysic.

Larry, the trouble with the text that you cite is that it overstates Husserl´s supposed naïvéte. Husserl never aspired to "eliminate the real world" but merely to place its presuppositions in abeyance. This is the meaning of epoché, and not suppression of spatiotemporal existence, despite Husserl´s own excessively negative formulae.

I totaly agree, Stefan. I myself see it as a reaction torwards Kant + the beginning of "Werturteilsstreit". And (I still defend this idea): The most important definition about what and how Husserl defines "strikte Wissenschaft" IS "Science, which suffices highest theoretical needs and allows with regard to ethical-religious a life that is regulated by pure rational norms."

Especially this genesis of norms (you said very well pointing at the fact that one has to  understand the zeitgeist) under the aspect of "with regard to" culture and "regulated by" the rational (both is a direct translation from the original german text; the translations of others may differ here) is very essential in order to understand what Husserl wants. If you take this as unimportant you won't be able to understand the text in a proper way. Because already in this sentence he declares borders and defines that theoretical science yes but not upon the cost of life. This is very important as it is also a milestone for ethical practise as we can see here an ethical imperativ already right from the start of this article.

Kant, himself (at least in the English translations I have read of his work) refers to "science" in ways we would not use it today.  At the time, at least to Kant, "science" was not limited to the scientific method.  To him, that would mean our use of only a posteriori knowledge to discover truth, which he agreed with Hume, would be impossible.  The problem of induction, Kant would say, becomes insurmountable when you let go of philosophy.  Science as we know it today, by it's own design, ignores a priori knowledge.  Einstein and Popper provided some excellent, more modern, insight on that question.